# AP Calculus AB : Integrals

## Example Questions

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### Example Question #21 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

A pot of water begins at a temperature of  and is heated at a rate of  degrees Celsius per minute. What will the temperature of the water be after 4 minutes?

Explanation:

Let  denote the temperature of the pot after  minutes.

The first thing to realize is that the quantity  is the derivative of . Using the fundamental theorem of Calculus, we know that

From there, we just need to solve the integral. Letting u = t+1, du=dt, we have the following:

Where the second equality follows by the power rule, and re-substituting t+1 = u.

Thus, we now have the equation . Because we know that the water started at , all we need to do is rearrange and substitute.

Yielding our final answer,

### Example Question #69 : Integrals

A ball is thrown into the air. It's height, after t seconds is modeled by the formula:

h(t)=-15t^2+30t feet.

At what time will the velocity equal zero?

0s

3s

5s

1.5s

1s

1s

Explanation:

In order to find where the velocity is equal to zero, take the derivative of the function and set it equal to zero.

h(t) = –15t+ 30t

h'(t) = –30t + 30

0 = –30t + 30

Then solve for "t".

–30 = –30t

t = 1

The velocity will be 0 at 1 second.