The velocity of a particle moving along the $x$-axis is $v(t) = 6\sqrt { t }- 2t$ units per second.

1. What was the average velocity over $0 ≤ t ≤ 15$ seconds?

   Step (a):

   $\frac{1}{15-0}\int_0^{15}\Big( 6\sqrt{t}-2t\Big)dt$

2. During the first $15$ seconds, what is the total distance the point traveled?

   Hint (b):

   $\int_0^{15}\Big|\Big( 6\sqrt{t}-2t\Big)\Big|dt$

   Graph the velocity function to see wether or not the point changes direction in the given interval. If it does, the integral needs to be broken into parts.

3. What was the particle’s total displacement over $0 ≤ t ≤ 15$ seconds?

   Step (c):

   $\int_0^{15}\Big( 6\sqrt{t}-2t\Big)dt$

4. What accounts for the large difference between the answers to parts (b) and (c)?