Question 4.4.4: Estimating Overall Change in Position An object moving along...
Estimating Overall Change in Position
An object moving along a straight line has velocity function v(t) = sin t. If the object starts at position 0, determine the total distance traveled and the object’s position at time t = 3π/2.
From the graph (see Figure 4.19), notice that sin t ≥ 0 for 0 ≤ t ≤ π and sin t ≤ 0 for π ≤ t ≤ 3π/2. The total distance traveled corresponds to the area of the shaded regions in Figure 4.19, given by
A=\int_0^\pi \sin t d t-\int_\pi^{3 \pi / 2} \sin t d t.
You can use the Midpoint Rule to get the following Riemann sums:
| n | R_n \approx \int_0^\pi \sin t d t | n | R_n \approx \int_\pi^{3 \pi / 2} \sin t d t |
| 10 | 2.0082 | 10 | -1.0010 |
| 20 | 2.0020 | 20 | -1.0003 |
| 50 | 2.0003 | 50 | -1.0000 |
| 100 | 2.0001 | 100 | -1.0000 |
Observe that the sums appear to be converging to 2 and −1, respectively, which we will soon be able to show are indeed correct. The total area bounded between y = sin t and the t-axis on \left[0, \frac{3 \pi}{2}\right] is then
\int_0^\pi \sin t d t-\int_\pi^{3 \pi / 2} \sin t d t=2+1=3,
so that the total distance traveled is 3 units. The overall change in position of the object is given by
\int_0^{3 \pi / 2} \sin t d t=\int_0^\pi \sin t d t+\int_\pi^{3 \pi / 2} \sin t d t=2+(-1)=1.
So, if the object starts at position 0, it ends up at position 0 + 1 = 1.
