# AP Calculus AB : Interpretations and properties of definite integrals

## Example Questions

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### Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

If f(1) = 12, f' is continuous, and the integral from 1 to 4 of f'(x)dx = 16, what is the value of f(4)?

28

12

4

27

16

28

Explanation:

You are provided f(1) and are told to find the value of f(4). By the FTC, the following follows:

(integral from 1 to 4 of f'(x)dx) + f(1) = f(4)

16 + 12 = 28

### Example Question #2 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Find the limit.

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n+ 6)

nonexistent

–6

1

4

0

4

Explanation:

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n+ 6)

Use L'Hopitals rule to find the limit.

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n+ 6)

lim as n approaches infiniti of ((12n2) – 6)/((3n2) – 4n + 6)

lim as n approaches infiniti of 24n/(6n – 4)

lim as n approaches infiniti of 24/6

The limit approaches 4.

### Example Question #3 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

If a particle's movement is represented by , then when is the velocity equal to zero?

Explanation:

now set  because that is what the question is asking for.

seconds

### Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

A particle's movement is represented by

At what time is the velocity at it's greatest?

Explanation:

The answer is at 6 seconds.

We can see that this equation will look like a upside down parabola so we know there will be only one maximum.

Now we set  to find the local maximum.

seconds

### 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 #8 : Calculus 3

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 #7 : 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 average value of the function  from  to ?

Explanation:

The average function value is given by the following formula:

, evaluated from  to .

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

If

then find .

Explanation:

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

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