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 \(0 \le t \le 4\text{,}\) where \(t\) is measured in minutes and \(v\) is measured in meters per minute.
(a)
Sketch an accurate graph of \(y = v(t)\text{.}\) What kind of curve is \(y = \sqrt{4-(t-2)^2}\text{?}\)
(b)
Evaluate \(\int_0^4 v(t) \, dt\) exactly.
(c)
In terms of the physical problem of the moving object with velocity \(v(t)\text{,}\) what is the meaning of \(\int_0^4 v(t) \, dt\text{?}\) Include units on your answer.
(d)
(e)
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 \(\int_0^4 v(t) \, dt\text{.}\) What is the rectangle’s exact height?
(f)
How can you use the average value you found in (d) to compute the total distance traveled by the moving object over \([0,4]\text{?}\)
