Solução_Calculo_Stewart_6e

F.
TX.10 SECTION 8.3 APPLICATIONS TO PHYSICS AND ENGINEERING ¤ 369
The computation of y in this problem (and many others) can be
simplified by using horizontal rather than vertical approximating rectangles.
If the length of a thin rectangle at coordinate y is (y), then its area is
(y) ∆y,itsmassisρ(y) ∆y, and its moment about the x-axis is
∆M x = ρy(y) ∆y. Thus,
M x = ρy(y) dy and y =
ρy(y) dy
ρA
= 1 y(y) dy
A
In this problem, (y) = c − a
b
(b − y) by similar triangles, so
y = 1 A
b
0
c − a
b
y(b − y) dy = 2 b 2 b
0 (by − y2 ) dy = 2 b 2 1
2 by2 − 1 3 y3 b
0 = 2 b 2 · b3 6 = b 3
Notice that only one integral is needed when this method is used.
41. Divide the lamina into two triangles and one rectangle with respective masses of 2, 2 and 4, so that the total mass is 8. Using
the result of Exercise 39, the triangles have centroids
−1, 2 3 and 1,
2
3 . The centroid of the rectangle (its center) is 0, −
1
2 .
So, using Formulas 5 and 7, we have y = M x
m = 1 3
m i y i = 1 8 2 2
3 +2 2
3 +4 −
1
2 =
1 2
8 3 =
1
12
,andx =0,
m
since the lamina is symmetric about the line x =0. Thus, the centroid is (x, y) = 0, 1 12
.
i=1
43.
b (cx + d) f(x) dx = b
cx f(x) dx + b
df(x) dx = c b
xf(x) dx + d b
f(x) dx = cxA + d b
a a a a a a
= cx b
f(x) dx + d b
f(x) dx =(cx + d) b
a a a
f(x) dx
f(x) dx [by (8)]
45. A cone of height h and radius r can be generated by rotating a right triangle
about one of its legs as shown. By Exercise 39, x = 1 3 r,sobytheTheoremof
Pappus, the volume of the cone is
V = Ad = 1
2 · base · height · (2πx) = 1 2 rh · 2π 1
3 r = 1 3 πr2 h.
47. Suppose the region lies between two curves y = f(x) and y = g(x) where f(x) ≥ g(x), as illustrated in Figure 13.
Choose points x i with a = x 0