James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 5.4 Indefinite Integrals and the Net Change Theorem 405
Figure 2 shows the graph of the integrand
in Example 4. We know from
Section 5.2 that the value of the integral
can be interpreted as a net area: the sum
of the areas labeled with a plus sign
minus the area labeled with a minus
sign.
y
3
2
Example 4 Find y S2x 3 2 6x 1 3 dx and interpret the result in terms of
areas.
0 x 2 1 1D
SOLUtion The Fundamental Theorem gives
y 2
S2x 3 2 6x 1 3 dx − 2
0 x 2 1 1D x 4
4 2 6 x 2
2
2 1 3 tan21 xG0
− 1 2 x 4 2 3x 2 1 3 tan 21 xg 2 0
− 1 2 s24 d 2 3s2 2 d 1 3 tan 21 2 2 0
− 24 1 3 tan 21 2
0
2 x
This is the exact value of the integral. If a decimal approximation is desired, we can use
a calculator to approximate tan 21 2. Doing so, we get
FIGURE 2
y 2
0
S2x 3 2 6x 1 3
x 2 1 1D dx < 20.67855
Example 5 Evaluate y 9 2t 2 1 t 2 st 2 1
dt.
1 t 2
SOLUtion First we need to write the integrand in a simpler form by carrying out the
division:
y 9
1
2t 2 1 t 2 st 2 1
dt − y 9
s2 1 t 1y2 2 t 22 d dt
t 2 1
n
− 2t 1 t 3y2
3
2
9
2 t21
− 2t 1 2 3 t 3y2 1 1 9
21G1
tG1
− (2 ? 9 1 2 3 ? 9 3y2 1 1 9) 2 (2 ? 1 1 2 3 ? 13y2 1 1 1)
− 18 1 18 1 1 9 2 2 2 2 3 2 1 − 32 4 9
n
Applications
Part 2 of the Fundamental Theorem says that if f is continuous on fa, bg, then
y b
f sxd dx − Fsbd 2 Fsad
a
where F is any antiderivative of f. This means that F9 − f , so the equation can be rewritten
as
y b
F9sxd dx − Fsbd 2 Fsad
a
We know that F9sxd represents the rate of change of y − Fsxd with respect to x and
Fsbd 2 Fsad is the change in y when x changes from a to b. [Note that y could, for
instance, increase, then decrease, then increase again. Although y might change in both
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