# Physics equations/19-Electric Potential and Electric Field/Q:SurfaceIntegralsCalculus/testbank

## c19ElectricPotentialField_SurfaceIntegral_v1

1 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.35+2.57z)\rho ^{3}{\hat {\rho }}+7.45z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 1.148E+03 b) 1.391E+03 c) 1.685E+03 d) 2.042E+03 e) 2.473E+03

2 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.05+2.59z)\rho ^{2}{\hat {\rho }}+7.4z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 6.908E+02 b) 8.369E+02 c) 1.014E+03 d) 1.228E+03 e) 1.488E+03

3 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.12+1.85z)\rho ^{3}{\hat {\rho }}+8.88z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 3.041E+02 b) 3.684E+02 c) 4.464E+02 d) 5.408E+02 e) 6.552E+02

4 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2+1.45z)\rho ^{2}{\hat {\rho }}+8.02z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 3.742E+02 b) 4.534E+02 c) 5.493E+02 d) 6.655E+02 e) 8.063E+02

5 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.14+2.8z)\rho ^{2}{\hat {\rho }}+9.94z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 2.810E+02 b) 3.404E+02 c) 4.124E+02 d) 4.996E+02 e) 6.053E+02

6 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(1.85+1.33z)\rho ^{3}{\hat {\rho }}+7.52z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 1.304E+03 b) 1.579E+03 c) 1.914E+03 d) 2.318E+03 e) 2.809E+03

7 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.07+2.87z)\rho ^{2}{\hat {\rho }}+9.56z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 7.933E+02 b) 9.611E+02 c) 1.164E+03 d) 1.411E+03 e) 1.709E+03

8 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.17+1.5z)\rho ^{2}{\hat {\rho }}+8.75z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 3.630E+02 b) 4.398E+02 c) 5.329E+02 d) 6.456E+02 e) 7.821E+02

9 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.28+1.72z)\rho ^{3}{\hat {\rho }}+7.33z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 2.597E+03 b) 3.147E+03 c) 3.812E+03 d) 4.619E+03 e) 5.596E+03

10 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.04+1.66z)\rho ^{2}{\hat {\rho }}+7.54z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 8.528E+02 b) 1.033E+03 c) 1.252E+03 d) 1.516E+03 e) 1.837E+03

11 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.21+1.16z)\rho ^{2}{\hat {\rho }}+7.96z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 3.417E+03 b) 4.140E+03 c) 5.016E+03 d) 6.077E+03 e) 7.362E+03

12 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.12+1.68z)\rho ^{2}{\hat {\rho }}+8.83z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 4.593E+03 b) 5.564E+03 c) 6.741E+03 d) 8.167E+03 e) 9.894E+03

13 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.05+2.05z)\rho ^{2}{\hat {\rho }}+9.62z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 4.489E+02 b) 5.438E+02 c) 6.589E+02 d) 7.983E+02 e) 9.671E+02

14 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(1.93+2.31z)\rho ^{3}{\hat {\rho }}+7.21z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 6.731E+02 b) 8.154E+02 c) 9.879E+02 d) 1.197E+03 e) 1.450E+03

15 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.24+1.11z)\rho ^{3}{\hat {\rho }}+8.16z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 2.769E+03 b) 3.354E+03 c) 4.064E+03 d) 4.923E+03 e) 5.965E+03

16 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(1.96+2.52z)\rho ^{2}{\hat {\rho }}+7.11z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 4.522E+02 b) 5.478E+02 c) 6.637E+02 d) 8.041E+02 e) 9.742E+02

17 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(1.86+2.43z)\rho ^{2}{\hat {\rho }}+9.75z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 6.201E+02 b) 7.513E+02 c) 9.102E+02 d) 1.103E+03 e) 1.336E+03

18 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.24+2.08z)\rho ^{2}{\hat {\rho }}+8.93z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 1.704E+03 b) 2.064E+03 c) 2.501E+03 d) 3.030E+03 e) 3.671E+03

19 A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(1.89+1.31z)\rho ^{3}{\hat {\rho }}+8.35z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 5.311E+02 b) 6.434E+02 c) 7.795E+02 d) 9.444E+02 e) 1.144E+03

20 A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.37+2.6z)\rho ^{2}{\hat {\rho }}+8.84z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 1.362E+03 b) 1.650E+03 c) 2.000E+03 d) 2.423E+03 e) 2.935E+03

21 A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.45+2.26z)\rho ^{2}{\hat {\rho }}+8.92z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 5.043E+02 b) 6.109E+02 c) 7.402E+02 d) 8.967E+02 e) 1.086E+03

22 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(1.88+1.29z)\rho ^{2}{\hat {\rho }}+7.2z^{2}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 1.248E+03 b) 1.512E+03 c) 1.832E+03 d) 2.220E+03 e) 2.689E+03

23 A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.44+2.86z)\rho ^{2}{\hat {\rho }}+7.42z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{top}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the top surface of the cylinder.

 a) 5.664E+03 b) 6.863E+03 c) 8.314E+03 d) 1.007E+04 e) 1.220E+04

## c19ElectricPotentialField_SurfaceIntegral_v1

A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.35+2.57z)\rho ^{3}{\hat {\rho }}+7.45z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\int _{side}{\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the curved side surface of the cylinder.

 a) 2.221E+03 b) 2.690E+03 c) 3.259E+03 d) 3.949E+03 e) 4.784E+03

copies
===2===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.05+2.59z)\rho^2\hat\rho +7.4z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 6.457E+02
-b) 7.823E+02
-c) 9.477E+02
-d) 1.148E+03
+e) 1.391E+03
===3===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.12+1.85z)\rho^3\hat\rho +8.88z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
+a) 8.525E+02
-b) 1.033E+03
-c) 1.251E+03
-d) 1.516E+03
-e) 1.837E+03
===4===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2+1.45z)\rho^2\hat\rho +8.02z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
+a) 4.021E+02
-b) 4.872E+02
-c) 5.902E+02
-d) 7.151E+02
-e) 8.663E+02
===5===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.14+2.8z)\rho^2\hat\rho +9.94z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 2.420E+02
-b) 2.931E+02
-c) 3.551E+02
+d) 4.303E+02
-e) 5.213E+02
===6===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.85+1.33z)\rho^3\hat\rho +7.52z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 2.622E+03
-b) 3.177E+03
-c) 3.849E+03
-d) 4.663E+03
+e) 5.649E+03
===7===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.07+2.87z)\rho^2\hat\rho +9.56z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
+a) 4.162E+02
-b) 5.042E+02
-c) 6.109E+02
-d) 7.401E+02
-e) 8.967E+02
===8===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.17+1.5z)\rho^2\hat\rho +8.75z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 2.454E+02
-b) 2.973E+02
-c) 3.601E+02
+d) 4.363E+02
-e) 5.286E+02
===9===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.28+1.72z)\rho^3\hat\rho +7.33z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 3.232E+03
-b) 3.915E+03
-c) 4.743E+03
-d) 5.747E+03
+e) 6.962E+03
===10===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.04+1.66z)\rho^2\hat\rho +7.54z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 9.431E+02
-b) 1.143E+03
+c) 1.384E+03
-d) 1.677E+03
-e) 2.032E+03
===11===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.21+1.16z)\rho^2\hat\rho +7.96z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 1.533E+03
-b) 1.857E+03
+c) 2.250E+03
-d) 2.725E+03
-e) 3.302E+03
===12===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.12+1.68z)\rho^2\hat\rho +8.83z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
+a) 2.158E+03
-b) 2.614E+03
-c) 3.167E+03
-d) 3.837E+03
-e) 4.649E+03
===13===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.05+2.05z)\rho^2\hat\rho +9.62z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 2.318E+02
-b) 2.808E+02
-c) 3.402E+02
+d) 4.122E+02
-e) 4.994E+02
===14===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.93+2.31z)\rho^3\hat\rho +7.21z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 6.546E+02
-b) 7.931E+02
-c) 9.609E+02
+d) 1.164E+03
-e) 1.410E+03
===15===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.24+1.11z)\rho^3\hat\rho +8.16z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 9.205E+02
-b) 1.115E+03
+c) 1.351E+03
-d) 1.637E+03
-e) 1.983E+03
===16===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.96+2.52z)\rho^2\hat\rho +7.11z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 4.027E+02
-b) 4.879E+02
+c) 5.911E+02
-d) 7.162E+02
-e) 8.676E+02
===17===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.86+2.43z)\rho^2\hat\rho +9.75z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
+a) 5.610E+02
-b) 6.796E+02
-c) 8.234E+02
-d) 9.975E+02
-e) 1.209E+03
===18===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.24+2.08z)\rho^2\hat\rho +8.93z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 3.799E+02
-b) 4.603E+02
-c) 5.576E+02
+d) 6.756E+02
-e) 8.185E+02
===19===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.89+1.31z)\rho^3\hat\rho +8.35z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 6.411E+02
-b) 7.767E+02
-c) 9.410E+02
+d) 1.140E+03
-e) 1.381E+03
===20===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.37+2.6z)\rho^2\hat\rho +8.84z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 7.465E+02
-b) 9.044E+02
-c) 1.096E+03
-d) 1.327E+03
+e) 1.608E+03
===21===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.45+2.26z)\rho^2\hat\rho +8.92z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 3.356E+02
-b) 4.066E+02
+c) 4.926E+02
-d) 5.968E+02
-e) 7.230E+02
===22===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.88+1.29z)\rho^2\hat\rho +7.2z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 1.579E+03
+b) 1.914E+03
-c) 2.318E+03
-d) 2.809E+03
-e) 3.403E+03
===23===
{<!--c19ElectricPotentialField_SurfaceIntegral_2-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.44+2.86z)\rho^2\hat\rho +7.42z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\int_{side}\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the curved side surface of the cylinder.}
-a) 1.692E+03
-b) 2.050E+03
+c) 2.484E+03
-d) 3.009E+03
-e) 3.645E+03


## c19ElectricPotentialField_SurfaceIntegral_v1

A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis. A vector field can be expressed in cylindrical coordinates as,
${\displaystyle {\vec {\mathfrak {F}}}=(2.35+2.57z)\rho ^{3}{\hat {\rho }}+7.45z^{3}{\hat {z}}}$
Let ${\displaystyle {\hat {n}}}$ be the outward unit normal to this cylinder and evaluate ,
${\displaystyle \left|\oint {\vec {\mathfrak {F}}}\cdot {\hat {n}}dA\right|\,}$
over the entire surface of the cylinder.

 a) 4.59E+03 b) 5.56E+03 c) 6.73E+03 d) 8.15E+03 e) 9.88E+03

copies
===2===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.05+2.59z)\rho^2\hat\rho +7.4z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 6.46E+02
-b) 7.82E+02
-c) 9.48E+02
-d) 1.15E+03
+e) 1.39E+03
===3===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.12+1.85z)\rho^3\hat\rho +8.88z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 3.96E+02
-b) 4.79E+02
-c) 5.81E+02
-d) 7.04E+02
+e) 8.53E+02
===4===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2+1.45z)\rho^2\hat\rho +8.02z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.13E+03
-b) 1.37E+03
-c) 1.66E+03
+d) 2.01E+03
-e) 2.44E+03
===5===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.14+2.8z)\rho^2\hat\rho +9.94z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 2.93E+02
-b) 3.55E+02
+c) 4.30E+02
-d) 5.21E+02
-e) 6.32E+02
===6===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.85+1.33z)\rho^3\hat\rho +7.52z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 3.18E+03
-b) 3.85E+03
-c) 4.66E+03
+d) 5.65E+03
-e) 6.84E+03
===7===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.07+2.87z)\rho^2\hat\rho +9.56z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.59E+03
-b) 1.93E+03
+c) 2.34E+03
-d) 2.83E+03
-e) 3.43E+03
===8===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.17+1.5z)\rho^2\hat\rho +8.75z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 3.60E+02
+b) 4.36E+02
-c) 5.29E+02
-d) 6.40E+02
-e) 7.76E+02
===9===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.28+1.72z)\rho^3\hat\rho +7.33z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.50E+04
+b) 1.82E+04
-c) 2.20E+04
-d) 2.66E+04
-e) 3.23E+04
===10===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.04+1.66z)\rho^2\hat\rho +7.54z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 9.43E+02
-b) 1.14E+03
+c) 1.38E+03
-d) 1.68E+03
-e) 2.03E+03
===11===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.21+1.16z)\rho^2\hat\rho +7.96z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 6.69E+03
-b) 8.10E+03
-c) 9.81E+03
-d) 1.19E+04
+e) 1.44E+04
===12===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.12+1.68z)\rho^2\hat\rho +8.83z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.29E+04
+b) 1.56E+04
-c) 1.89E+04
-d) 2.30E+04
-e) 2.78E+04
===13===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.05+2.05z)\rho^2\hat\rho +9.62z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.09E+03
-b) 1.32E+03
-c) 1.60E+03
-d) 1.94E+03
+e) 2.35E+03
===14===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.93+2.31z)\rho^3\hat\rho +7.21z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 5.40E+02
-b) 6.55E+02
-c) 7.93E+02
-d) 9.61E+02
+e) 1.16E+03
===15===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.24+1.11z)\rho^3\hat\rho +8.16z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 4.69E+03
-b) 5.69E+03
+c) 6.89E+03
-d) 8.35E+03
-e) 1.01E+04
===16===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.96+2.52z)\rho^2\hat\rho +7.11z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
+a) 5.91E+02
-b) 7.16E+02
-c) 8.68E+02
-d) 1.05E+03
-e) 1.27E+03
===17===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.86+2.43z)\rho^2\hat\rho +9.75z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 4.63E+02
+b) 5.61E+02
-c) 6.80E+02
-d) 8.23E+02
-e) 9.98E+02
===18===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.24+2.08z)\rho^2\hat\rho +8.93z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 3.13E+03
-b) 3.79E+03
-c) 4.59E+03
-d) 5.56E+03
+e) 6.74E+03
===19===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.89+1.31z)\rho^3\hat\rho +8.35z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 9.41E+02
+b) 1.14E+03
-c) 1.38E+03
-d) 1.67E+03
-e) 2.03E+03
===20===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.37+2.6z)\rho^2\hat\rho +8.84z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 4.63E+03
+b) 5.61E+03
-c) 6.79E+03
-d) 8.23E+03
-e) 9.97E+03
===21===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=2, and height, h=4, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.45+2.26z)\rho^2\hat\rho +8.92z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.29E+03
-b) 1.56E+03
-c) 1.89E+03
+d) 2.29E+03
-e) 2.77E+03
===22===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (1.88+1.29z)\rho^2\hat\rho +7.2z^2\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 1.08E+03
-b) 1.30E+03
-c) 1.58E+03
+d) 1.91E+03
-e) 2.32E+03
===23===
{<!--c19ElectricPotentialField_SurfaceIntegral_3-->A cylinder of radius, r=3, and height, h=6, is centered at the origin and oriented along the z axis.  A vector field can be expressed in cylindrical coordinates as,  <br> $\vec\mathfrak F = (2.44+2.86z)\rho^2\hat\rho +7.42z^3\hat z$<br> Let $\hat n$ be the outward unit normal to this cylinder and evaluate ,<br> $\left |\oint\vec\mathfrak F\cdot\hat n dA\right|\,$<br>over the entire surface of the cylinder.}
-a) 9.41E+03
-b) 1.14E+04
+c) 1.38E+04
-d) 1.67E+04
-e) 2.03E+04