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
We will need the following theorem in the discussion of the Fundamental Theorem of Calculus.
Suppose is continuous on . Then for some .
Proof of the Mean Value Theorem for Integration
satisfies the requirements of the Extreme Value Theorem, so it has a minimum and a maximum in . Since
- for all
Since is continuous, by the Intermediate Value Theorem there is some with such that
Fundamental Theorem of Calculus
Statement of the Fundamental Theorem
Suppose that is continuous on . We can define a function by
Suppose is continuous on and is defined by
Then is differentiable on and for all ,
When we have such functions and where for every in some interval we say that is the antiderivative of on .
Suppose that is continuous on and that is any antiderivative of .
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
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
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
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
Integration of Polynomials
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.
- Power Rule of Integration I
- as long as and .
Notice that we allow all values of , even negative or fractional. If then this works even if includes .
- Power Rule of Integration II
- as long as .
- 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
The Fundamental Theorem of Calculus, Part 1
The Fundamental Theorem of Calculus, Part 2
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