AP Calculus BC : Fundamental Theorem of Calculus with Definite Integrals

Example Questions

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Example Question #1 : Fundamental Theorem Of Calculus With Definite Integrals

Find the result:

Explanation:

Set . Then , and by the chain rule,

By the fundamental theorem of Calculus, the above can be rewritten as

Example Question #2 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate :

Explanation:

By the Fundamental Theorem of Calculus, we have that . Thus,

Example Question #3 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate  when .

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function.

Therefore .

Example Question #1 : Fundamental Theorem Of Calculus With Definite Integrals

Evaluate  when .

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function . Therefore, .

Example Question #5 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Suppose we have the function

What is the derivative, ?

Explanation:

We can view the function  as a function of , as so

where .

We can find the derivative of  using the chain rule:

where  can be found using the fundamental theorem of calculus:

So we get

Example Question #6 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Given

, what is ?

None of the above.

Explanation:

By the Fundamental Theorem of Calculus, for all functions  that are continuously defined on the interval  with  in  and for all functions  defined by by , we know that .

Thus, for

.

Therefore,

Example Question #7 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Given

, what is ?

None of the above.

Explanation:

By the Fundamental Theorem of Calculus,  for all functions  that are continuously defined on the interval  with  in  and for all functions  defined by by , we know that .

Given

, then

.

Therefore,

.

Example Question #8 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate

Explanation:

Use the fundamental theorem of calculus to evaluate:

Example Question #9 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Explanation:

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints: