# AP Calculus AB : Integrals

## Example Questions

### Example Question #5 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

What is the domain of ?

Explanation:

because the denominator cannot be zero and square roots cannot be taken of negative numbers

### Example Question #6 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

If ,

then at , what is 's instantaneous rate of change?

Explanation:

### Example Question #1 : Interpretations And Properties Of Definite Integrals

Which of the following represents the graph of the polar function  in Cartestian coordinates?

Explanation:

First, mulitply both sides by r.

Then, use the identities  and .

### Example Question #1 : Interpretations And Properties Of Definite Integrals

What is the average value of the function  from  to ?

Explanation:

The average function value is given by the following formula:

, evaluated from  to .

### Example Question #2 : Interpretations And Properties Of Definite Integrals

If

then find .

Explanation:

We see the answer is 0 after we do the quotient rule.

### Example Question #51 : Integrals

If , then which of the following is equal to ?

Explanation:

According to the Fundamental Theorem of Calculus, if we take the derivative of the integral of a function, the result is the original function. This is because differentiation and integration are inverse operations.

For example, if , where  is a constant, then .

We will apply the same principle to this problem. Because the integral is evaluated from 0 to , we must apply the chain rule.

Explanation:

Explanation: