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.
As an illustrative example see § 1.8 for the connection of natural logarithm and 1/x.
Suppose that is continuous on . We can define a function by
Fundamental Theorem of Calculus Part I
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 .
Fundamental Theorem of Calculus Part II
Suppose that is continuous on and that is any antiderivative of .
Then
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.
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
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 .
Examples
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