Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 5 REVIEW ¤ 255
5 Review
1. (a) n
i=1 f(x∗ i ) ∆x is an expression for a Riemann sum of a function f.
x ∗ i is a point in the ith subinterval [x i−1 ,x i ] and ∆x is the length of the subintervals.
(b)SeeFigure1inSection5.2.
(c) In Section 5.2, see Figure 3 and the paragraph beside it.
2. (a) See Definition 5.2.2.
(b)SeeFigure2inSection5.2.
(c) In Section 5.2, see Figure 4 and the paragraph by it (contains “net area”).
3. See the Fundamental Theorem of Calculus after Example 9 in Section 5.3.
4. (a) See the Net Change Theorem after Example 5 in Section 5.4.
(b) t 2
t 1
r(t) dt represents the change in the amount of water in the reservoir between time t 1 and time t 2.
5. (a) 120
60
v(t) dt represents the change in position of the particle from t =60to t =120seconds.
(b) 120
60
|v(t)| dt represents the total distance traveled by the particle from t =60to 120 seconds.
(c) 120
60
a(t) dt represents the change in the velocity of the particle from t =60to t =120seconds.
6. (a) f(x) dx is the family of functions {F | F 0 = f}. Any two such functions differ by a constant.
(b) The connection is given by the Net Change Theorem: b
f(x) dx = f(x) dx b
if f is continuous.
a a
7. The precise version of this statement is given by the Fundamental Theorem of Calculus. See the statement of this theorem and
the paragraph that follows it at the end of Section 5.3.
8. See the Substitution Rule (5.5.4). This says that it is permissible to operate with the dx afteranintegralsignasifitwerea
differential.
1. True by Property 2 of the Integral in Section 5.2.
3. True by Property 3 of the Integral in Section 5.2.
5. False. For example, let f(x) =x 2 .Then 1
√
x2 dx =
1
xdx= 1 ,but 1
0
0 2 0 x2 1
dx = = √ 1
3 3
.
7. True by Comparison Property 7 of the Integral in Section 5.2.
9. True. The integrand is an odd function that is continuous on [−1, 1], so the result follows from Theorem 5.5.7(b).