Calculus 2 : Introduction to Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #161 : Integrals

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Possible Answers:

Correct answer:

Explanation:

The antiderivative of   is  .

Evaluating  (by the fundamental theorem of calculus) gives us...

Example Question #162 : Integrals

Solve

Possible Answers:

Correct answer:

Explanation:

The antiderivative of   is  .

Evaluating  (by the fundamental theorem of calculus) gives us...

 

Example Question #163 : Integrals

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Possible Answers:

Correct answer:

Explanation:

The antiderivative of  is .

By evaluating  (by the fundamental theorem of calculus) we get...

 

Example Question #164 : Integrals

Evaluate the definite integral using the Fundamental Theorem of Calculus.

Possible Answers:

Correct answer:

Explanation:

The antiderivative of  is .

By evaluating  (by the fundamental theorem of calculus) we get...

 

Example Question #165 : Integrals

Given , what is ?

Possible Answers:

Correct answer:

Explanation:

According to the Fundamental Theorem of Calculus, if  is a continuous function on the interval  with  as the function defined for all  on  as 

, then .

Therefore, if 

, then 

.

Thus, 

.

Example Question #166 : Integrals

Given , what is ?

Possible Answers:

Correct answer:

Explanation:

According to the Fundamental Theorem of Calculus, if  is a continuous function on the interval  with  as the function defined for all  on  as 

, then .

Therefore, if 

, then 

.

Thus, 

.

Example Question #167 : Integrals

Given , what is ?

Possible Answers:

Correct answer:

Explanation:

According to the Fundamental Theorem of Calculus, if  is a continuous function on the interval  with  as the function defined for all  on  as 

, then .

Therefore, if 

.

Thus, 

.

Example Question #168 : Integrals

Write  in integral form, if  is position and  where  is velocity at time 

Possible Answers:

Correct answer:

Explanation:

To write position in integral form, we can take advantage of the fundamental theorem of calculus. Since the bounds are  and , and 

 

Example Question #169 : Integrals

Given that , determine:

Possible Answers:

Correct answer:

Explanation:

Since , we know that

By the fundamental theorem of calculus:

Example Question #170 : Integrals

Find  of 

Possible Answers:

Correct answer:

Explanation:

This is a Second Fundamental Theorem of Calculus.  Since derivatives and anti-derivatives annihilate each other, we simply need to plug in the bounds into the function and multiply each by their derivative, respectively.

 

The second term drops out since the derivative of zero is zero.

In the first term, we again have two functions that annihilate each other:

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