Activity 4.3.4

Suppose that $v(t) = \sqrt{4-(t-2)^2}$ tells us the instantaneous velocity of a moving object on the interval $0 \le t \le 4$, where $t$ is measured in minutes and $v$ is measured in meters per minute.

1. Sketch an accurate graph of $y = v(t)$. What kind of curve is $y = \sqrt{4-(t-2)^2}$?

2. Evaluate $\int_0^4 v(t) \, dt$ exactly.

3. In terms of the physical problem of the moving object with velocity $v(t)$, what is the meaning of $\int_0^4 v(t) \, dt$? Include units on your answer.

4. Determine the exact average value of $v(t)$ on $[0,4]$. Include units on your answer.

5. Sketch a rectangle whose base is the line segment from $t=0$ to $t = 4$ on the $t$-axis such that the rectangle's area is equal to the value of $\int_0^4 v(t) \, dt$. What is the rectangle's exact height?

6. How can you use the average value you found in (d) to compute the total distance traveled by the moving object over $[0,4]$?