Math 230 Quiz 2 Solutions Spring 2003
1. Find the number bsuch that the line y=bdivides the region bounded by the
curves y=x2and y= 4 into two regions with equal area.
Z4
0
xdy = 2 Zb
0
xdy =⇒Z4
0
y1/2dy = 2 Zb
0
y1/2dy
=⇒2
3hy3/2i4
0= 22
3hy3/2ib
0
=⇒2
3(8 −0) = 4
3(b3/2−0) =⇒b3/2=6
128 = 48
12 = 4 =⇒b= 42/3
Or: Z2
0
(4 −x2)dx = 2 Z√b
0
(4 −x2)dx
=⇒4x−1
3x3|2
0= 2(4x−1
3x3)|√b
0
=⇒8−8
3= 2(4√b−1
3b3/2) =⇒16
3= 8b1/2−2
3b3/2
2. Find the volume of the solid obtained by rotating the region bounded by y=x,
y= 0, x= 2, x= 4 about the line x= 1.
For 0 ≤y < 2, a cross-section is an annulus with inner radius (2 −1) and outer
radius (4 −1), the area of which is
A1(y) = π(4 −1)2−π(2 −1)2
For 2 ≤y≤4 a cross-section is an annulus with inner radius y−1 and outer radius
(4 −1), the area of which is
A2(y) = π(4 −1)2−π(y−1)2
=⇒V=πZ2
0
[(4 −1)2−(2 −1)2]dy +πZ4
2
[(4 −1)2−(y−1)2]dy
=π[8y]2
0+πZ4
2
(8 + 2y−y2)dy = 16π+π8y+y2−1
3y34
2
16π+π32 + 16 −64
3−16 + 4 −8
3 =76π
3
Or: Z4
2
2π(x−1)xdx =Z4
2
2π(x2−x)dx = (2π1
3x3−1
2x2)|4
2=64
3−8−8
3−22π
=56
3−62π=38
32π=76π
3
3. A force of 10 lb. is required to hold a spring stretched 4 in. beyond its natural
length. How much work is done in stretching it from its natural length to 6 in.
beyond its natural length?
10 = f(x) = kx =1
3k(4 inches = 1/3 foot), so k= 30lb/ft and f(x) = 30x. Now
6 inches = 1/2 foot, so W=Z1/2
0
30xdx = [15x2]1/2
0=15
4ft-lb.