Definite integral

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Definite Intergral[edit]

Integration is a mathematic operation on fuction to find area under the function curve bounded by region from a to b The integral with respect to x of a real-valued function f(x) of a real variable x on the interval [a, b] is written as

The integral sign represents integration.
The symbol dx, called the differential of the variable x, indicates that the variable of integration is x.
The function f(x) to be integrated is called the integrand.
The symbol dx is separated from the integrand by a space (as shown). If a function has an integral, it is said to be integrable. The points a and b are called the limits of the integral. An integral where the limits are specified is called a definite integral.
The integral is said to be over the interval [a, b].

The fundamental theorem of calculus is a critical portion of calculus because it links the concept of a derivative to that of an integral. As a result, we can use our knowledge of derivatives to find the area under the curve, which is often quicker and simpler than using the definition of the integral.

Mean Value Theorem for Integration[edit]

We will need the following theorem in the discussion of the Fundamental Theorem of Calculus.


Proof of the Mean Value Theorem for Integration[edit]

satisfies the requirements of the Extreme Value Theorem, so it has a minimum and a maximum in . Since

and since

for all

we have

Since is continuous, by the Intermediate Value Theorem there is some with such that

Fundamental Theorem of Calculus[edit]

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Statement of the Fundamental Theorem[edit]

Suppose that is continuous on . We can define a function by


When we have such functions and where for every in some interval we say that is the antiderivative of on .


Figure 1

Note: a minority of mathematicians refer to part one as two and part two as one. All mathematicians refer to what is stated here as part 2 as The Fundamental Theorem of Calculus.


Proof of Fundamental Theorem of Calculus Part I[edit]

Suppose . Pick so that . Then


Subtracting the two equations gives


so rearranging this we have

According to the Mean Value Theorem for Integration, there exists a such that

Notice that depends on . Anyway what we have shown is that

and dividing both sides by gives

Take the limit as we get the definition of the derivative of at so we have

To find the other limit, we will use the squeeze theorem. , so . Hence,

As is continuous we have

which completes the proof.

Proof of Fundamental Theorem of Calculus Part II[edit]

Define . Then by the Fundamental Theorem of Calculus part I we know that is differentiable on and for all

So is an antiderivative of . Since we were assuming that was also an antiderivative for all ,

Let . The Mean Value Theorem applied to on with says that

for some in . But since for all in , must equal for all in , i.e. g(x) is constant on .

This implies there is a constant such that for all ,

and as is continuous we see this holds when and as well. And putting gives

Notation for Evaluating Definite Integrals[edit]

The second part of the Fundamental Theorem of Calculus gives us a way to calculate definite integrals. Just find an antiderivative of the integrand, and subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound. That is

where . As a convenience, we use the notation

to represent

Integration of Polynomials[edit]

Using the power rule for differentiation we can find a formula for the integral of a power using the Fundamental Theorem of Calculus. Let . We want to find an antiderivative for . Since the differentiation rule for powers lowers the power by 1 we have that

As long as we can divide by to get

So the function is an antiderivative of . If then is continuous on and, by applying the Fundamental Theorem of Calculus, we can calculate the integral of to get the following rule.


Notice that we allow all values of , even negative or fractional. If then this works even if includes .


  • To find we raise the power by 1 and have to divide by 4. So
  • The power rule also works for negative powers. For instance
  • We can also use the power rule for fractional powers. For instance
  • Using linearity the power rule can also be thought of as applying to constants. For example,
  • Using the linearity rule we can now integrate any polynomial. For example