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Mathematical Expression Editor

In this section we learn the second part of the fundamental theorem and we use it to
compute the derivative of an area function.

The Fundamental Theorem of Calculus, Part II

Area Functions

Area Function Let be a continuous function on the interval and let be a
number in the interval. Then, an area function is a function of the form

An area function gives us the net area between the curve and the -axis over the
interval . Net area means the area under the curve (where the function, , is positive)
minus the area above the curve (where the function, , is negative).

Use the graph of to find the values of the area function

Use the graph of to find the values of the area function

From to the curve is a semi-circle

FTC, Part II

The second part of the FTC tells us the derivative of an area function .

Fundamental Theorem of Calculus, Part II If is continuous on the closed interval
then for any value of in the interval .

This conclusion establishes the theory of the existence of anti-derivatives, i.e.,
thanks to the FTC, part II, we know that every continuous function has an
anti-derivative.

Let . This function is continuous for all . Consider the function By the FTC part II
we can say that for any number and any value of , This example asserts that the
continuous function has an anti-derivative. Finding another form- a familiar form- is
an impossible task, despite the simple nature of the function itself. Thus the
theory of anti-differentiation is much richer than the theory of differentiation,
since such a simple function can be differentiated easily (using the chain
rule).

Find the derivative of the function

By the FTC part II,

Note that this equation can be verified easily by computing the integral and then
taking the derivative of the result: and

Find the derivative of the function

By the FTC part II,

Find the derivative of the function First, we rewrite the integral as Then, by the
FTC part II,

Find the following derivative: To solve this problem, we will need to use the chain
rule. Let Then the function we wish to differentiate is . By the chain rule, By FTC,
part II, and substituting for , we have Now multiplying by gives the final answer:

Find the following derivative: To solve this problem, we will need to use the chain
rule. Let Then the function we wish to differentiate is . Note that the negative sign
comes from switching the endpoints of integration. By the chain rule, By FTC, part
II, and substituting for , we have Now multiplying by gives the final answer:

Here are some detailed, lecture style videos on area functions: