Suppose that an object moving along a straight line path has its velocity in feet per second at time \(t\) in seconds given by \(v(t) = \frac{2}{9}(t-3)^2 + 2\text{.}\)
Carefully sketch the region whose exact area will tell you the value of the distance the object traveled on the time interval \(2 \le t \le 5\text{.}\)
Compute \(M_5\text{,}\) the middle Riemann sum, for \(v\) on the time interval \([1,5]\text{.}\) Be sure to clearly identify the value of \(\Delta t\) as well as the locations of \(t_0\text{,}\)\(t_1\text{,}\)\(\cdots\text{,}\)\(t_5\text{.}\) In addition, provide a careful sketch of the function and the corresponding rectangles that are being used in the sum.
Use appropriate computing technology such as this applet to compute \(M_{10}\) and \(M_{20}\text{.}\) What exact value do you think the middle sum eventually approaches as \(n\) increases without bound? What does that number represent in the physical context of the overall problem?
