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## Homework Statement

The rate of change of the altutide of a hot air balloon is given by r(t)= t

^{3}- 4t

^{2}+ 6 for 0 ≤ t ≤ 8. Which of the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?

a. ∫r(t)dt when t goes from 1.572 to 3.517

b. ∫r(t)dt when t goes from 0 to 8

c. ∫r(t)dt when t goes from 0 to 2.667

d. ∫r'(t)dt when t goes from 1.572 to 3.514

e. ∫r'(t)dt when t goes from 0 to 2.667

## Homework Equations

## The Attempt at a Solution

All I know is that "the rate of change" means the derivatve and when the altitude is decreasing, the answer should be negative.

But besides not knowing how to start this problem, I'm a little confused with all of the answer choices.

For answer d, the integral and derivative cancel each other out. If that's the case, does that mean the only thing you have to do is plug in the t values into r(t), like so?

∫r'(t)dt when t goes from 1.572 to 3.514

r(t) when t goes from 1.572 to 3.514

r(t)= t

r(t)= {3.514

r(t) when t goes from 1.572 to 3.514

r(t)= t

^{3}- 4t^{2}+ 6r(t)= {3.514

^{3}- 43.514^{2}+ 6} - {1.572^{3}- 41.572^{2}+ 6}?