Help with math, please anyone

Question

A particle moves along the x-axis with velocity v(t) = t² - 1, with t measured in seconds and v(t) measured in feet per second. Find the total distance travelled by the particle from t = 0 to t = 2 seconds.

Options:

1)	0.667
2)	2
3)	4
4)	None of these

Answer 1

v(t) = t² - 1

1) Find out if this graph intersects zero at any point between t = 0 and t = 2.

t² - 1 = 0

(t + 1)(t - 1) = 0

t = -1, 1. But we’re only concerned about positive values for t, so we’re focused on t = 1. 

2) Take the antiderivative of v(t), and evaluate it with upper bound 1 and lower bound 0. 

∫(t² - 1) dt

= t³ ∕ 3 - t

When we plug in 1, we get: -2 / 3.

When we plug in 0, we get: 0

-2 / 3 - 0 = -2 / 3. 

3) Take the antiderivative of v(t), and evaluate it with upper bound 2 and lower bound 1. 

∫(t² - 1) dt

= t³ ∕ 3 - t

When we plug in 2, we get: 2 / 3.

When we plug in 1, we get: - 2 / 3.

(2 / 3) - (-2 / 3) = 4 / 3

4) Add the result from 2) to the result from 3). So:

(4 / 3) + (-2 / 3) = 2 / 3.

This makes the answer 2 / 3, or 0.667.