Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Section 7.1 Integral as Net Change 381<br />
(a) Figure 7.2a supports that the position of the particle at t 1 is 163.<br />
(b) Figure 7.2b shows the position of the particle is 44 at t 5. Therefore, the displacement<br />
is 44 9 35.<br />
Now try Exercise 1(b).<br />
T = 1<br />
X = 5.3333333 Y = 1<br />
[–10, 50] by [–2, 6]<br />
(a)<br />
T = 5<br />
X = 44 Y = 5<br />
[–10, 50] by [–2, 6]<br />
(b)<br />
Figure 7.2 Using TRACE and the<br />
parametrization in Example 2 you can<br />
“see” the left and right motion of the<br />
particle.<br />
The reason for our method in Example 2 was to illustrate the modeling step that will<br />
be used throughout this chapter. We can also solve Example 2 using the techniques of<br />
Chapter 6 as shown in Exploration 1.<br />
EXPLORATION 1<br />
Revisiting Example 2<br />
The velocity of a particle moving along a horizontal s-axis for 0 t 5 is<br />
d s<br />
t<br />
dt<br />
2 .<br />
t <br />
81 2<br />
1. Use the indefinite integral of dsdt to find the solution of the initial value<br />
problem<br />
d s<br />
t<br />
dt<br />
2 , t <br />
81 2 s0 9.<br />
2. Determine the position of the particle at t 1. Compare your answer with the<br />
answer to Example 2a.<br />
3. Determine the position of the particle at t 5. Compare your answer with the<br />
answer to Example 2b.<br />
We know now that the particle in Example 1 was at s0 9 at the beginning of the<br />
motion and at s5 44 at the end. But it did not travel from 9 to 44 directly—it began its<br />
trip by moving to the left (Figure 7.2). How much distance did the particle actually travel?<br />
We find out in Example 3.<br />
EXAMPLE 3<br />
Calculating Total Distance Traveled<br />
Find the total distance traveled by the particle in Example 1.<br />
SOLUTION<br />
Solve Analytically We partition the time interval as in Example 2 but record every<br />
position shift as positive by taking absolute values. The Riemann sum approximating<br />
total distance traveled is<br />
vt k Δt,<br />
and we are led to the integral<br />
Total distance traveled 5<br />
vt dt 5<br />
8<br />
0<br />
0 t2 t 1 2 dt.<br />
Evaluate Numerically We have<br />
NINT ( t 2 t <br />
81 2 , t, 0,5) 42.59. Now try Exercise 1(c).