Chapter 6.1, Problem 36E

To determine

The value of $y=v(t)$ between 3 and 15.

Answer to Problem 36E

The velocity is constant over interval then area is rectangular already then it gives actual value.

Explanation of Solution

Given information:

The truck moves with the positive velocity v (t) from time t = 3 to time t = 15. The area under the graph of $y=v(t)$ between 3 and 5.

Correct option is (D).

The area under the graph of function is the total distance travelled between given time interval. Because, when the time interval is being partitioned into many tiny subintervals, each one so small that the velocity it would be essential constant. For MRAM method draw rectangles at midpoint of sub-intervals, height is given by velocity at that point.

Now

  $\begin{array}{l}\text{area under curve = area of rectage 1+}\mathrm{.....}\text{+area of last rectangle}\\ \text{=}\left(\text{height}\right)\left(\text{length of sub-interval}\right)+\mathrm{.....}+\left(\text{height}\right)\left(\text{length}\text{of sub-interval}\right)\\ =\text{distance travelved during sub-interval+}\mathrm{....}\text{+distance traveled during sub-interval}\\ \text{area under curve =total distance traveled during given interval}\end{array}$

Midpoint rectangular approximation method gives close to actual value if velocity is varying; if velocity is constant over interval then area is rectangular already then it gives actual value.

Conclusion:

The velocity is constant over interval then area is rectangular already then it gives actual value.