# Calculus II/Test 2

collapsed with cot and cob

### Chap 6.1 Applications of integration: More about areas

Chapter 6.1 Examples 1, 2, 3 (pp.432-4)

Also look at Examples 4 and 5 (only examples with the pencil icon are likely to appear on the test).

#### Matlab exercise (not on test)

Explain (with sketches) you you find the area enclosed by the following curves:

${\displaystyle y=\ln x^{2},\quad xy=3,\quad x=3,\quad x=4}$

MATLAB code (discussed in class)
clear all; close all; clc;
x = linspace(.5,10)
y = log(x.^2);
plot(x,y,'-');hold on;
y = 3./x;
plot(x,y,'-');
x(1:100)=3
y=linspace(-5,5)
plot(x,y,'-');hold on;
x(1:100)=4
y=linspace(-5,5);hold on;
plot(x,y,'-');


### Chapter 6.2 Applications of integration: Volumes

Chapter 6.2: Examples 1, 2, 3, and 4 (page 438) ---26 January 2017

### Chapter 6.3 Applications of integration: Volumes by cylindrical shells

Chapter 6.2: Examples 1 -- 18:24, 13 February 2017 (UTC) (UTC)

### Chapter 6.4 Applications of integration: Arc length

#### ? Place this on test as extra credit?

Here, Δs2≈Δx2+Δy2 is only an approximation because the segments are not infinitesimally small

Why Wikipedia (though great) is not enough: w:Special:Permalink/744766508#Derivation documents an important derivation of w:Arc length that was added in 2006. First,it is not sufficiently complete for introductory students, and second, it was removed as can be seen in this 2017 version of the article: w:Special:Permalink/754122488

${\displaystyle ds={\sqrt {dx^{2}+dy^{2}}}}$ leads both:
${\displaystyle {\frac {ds}{dt}}={\sqrt {{\frac {dx^{2}}{dt^{2}}}+{\frac {dy^{2}}{dt^{2}}}}}\Rightarrow {\frac {ds}{dx}}={\sqrt {1+\left(y\,'\right)^{2}}}}$

In the last step we replaced t by x. Now integrate WRT t or x. to get the desired result---26 January 2017

Infinitesimal is an interesting article, but not useful for this course.

### floating illusion (not on test)

click to expand (not on test)
fullFileName = 'C:\Users\Zach\Pictures\GIFs\AnimatedGifs\electricity.gif';
[gifImage cmap] = imread(fullFileName, 'Frames', 'all');
size(gifImage); implay(gifImage);


### Chap 6.5 Applications of integration: Average value of a function

Examples 1 and 3 (For possible extra credit study the proof above p461) 10,13 February 2017 (UTC)

### Problems CERTAIN to be on Test II

On Monday and Wednesday, I will will spend some time on questions that will certainly be on the test--Guy vandegrift (discusscontribs) Monday, 6 March 2017 (UTC)

Area between curves EXAMPLE 2 Section 6.1 p.433. I like this question because it emphasizes that an integral is a Reimann Sum, here of rectangles.

Using the washer method EXAMPLE 4 Section 6.2 p.442. Here, the integral is a Riemann sum of "little volumes", not "little rectangles".

NOT IN BOOK BUT ON TEST: Use the known circumference of the unit circle to generate an expression for a definite integral from x=0 to x=1. Do not solve the integral, but someone needs to verify that it is correct:

${\displaystyle \int _{0}^{1}{\sqrt {(1+(f')^{2}}}dx}$ where ${\displaystyle y=f(x)}$ and ${\displaystyle x^{2}+y^{2}=1\Rightarrow \int _{0}^{1}{\frac {dx}{\sqrt {1+x^{2}}}}={\frac {\pi }{2}}}$
(extra credit: explain why this is the formula for the arclength ${\displaystyle \left(\int d\ell =\int {\sqrt {dx^{2}+dy^{2}}}\right)}$).