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

408 Chapter 5 Integrals
Example 7 Figure 4 shows the power consumption in the city of San Francisco for
a day in September (P is measured in megawatts; t is measured in hours starting at
midnight). Estimate the energy used on that day.
P
800
600
400
200
FIGURE 4
0 3 6 9 12 15 18 21 t
Pacific Gas & Electric
SOLUTION Power is the rate of change of energy: Pstd − E9std. So, by the Net Change
Theorem,
y 24
0
Pstd dt − y 24
E9std dt − Es24d 2 Es0d
0
is the total amount of energy used on that day. We approximate the value of the integral
using the Midpoint Rule with 12 subintervals and Dt − 2:
y 24
Pstd dt < fPs1d 1 Ps3d 1 Ps5d 1 ∙ ∙ ∙ 1 Ps21d 1 Ps23dg Dt
0
< s440 1 400 1 420 1 620 1 790 1 840 1 850
− 15,840
1 840 1 810 1 690 1 670 1 550ds2d
The energy used was approximately 15,840 megawatt-hours.
n
A note on units
How did we know what units to use for energy in Example 7? The integral y 24
0
Pstd dt
is defined as the limit of sums of terms of the form Pst i *d Dt. Now Pst i *d is measured in
megawatts and Dt is measured in hours, so their product is measured in megawatt-hours.
The same is true of the limit. In general, the unit of measurement for y b a
f sxd dx is the
product of the unit for f sxd and the unit for x.
1–4 Verify by differentiation that the formula is correct.
1. y
1
dx − 2 s1 1 x 2
x 2 s1 1 x 2 x
2. y cos 2 x dx − 1 2 x 1 1 4 sin 2x 1 C
3. y tan 2 x dx − tan x 2 x 1 C
1 C
4. y xsa 1 bx dx − 2
15b 2 s3bx 2 2adsa 1 bxd3y2 1 C
5–18 Find the general indefinite integral.
5. y sx 1.3 1 7x 2.5 d dx
6. y s 4 x 5 dx
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