SolnEx1S14.nb
H* Find the average value of fHxL=ln xx on @1, eD*L
AVEF = Integrate@Log@xD  x, 8x, 1, E<D  HE - 1L
1
2 H- 1 + ãL
H* Find the arclength of y=2x32 0£x£7 *L
ArcLength = Integrate@Sqrt@9 x + 1D, 8x, 0, 7<D
1022
27
H* Find the area of the surface generated by revolving about the x axis the curve y=x^5,
2£x£3L
*L
SurfaceArea = 2 Pi Integrate@x ^ 5 Sqrt@1 + H5 x ^ 4L ^ 2D, 8x, 2, 3<D
1
Π - 192
6401 + 2187
164 026 +
15
1 3 7
1 3 7
1458 Hypergeometric2F1 B , , , - 164 025F - 128 Hypergeometric2F1 B , , , - 6400F
2 4 4
2 4 4
H*Find the area of the region enclosed by y=x^2, y=x^4, x=0 and x=2. *L
Area1 = Integrate@x ^ 2 - x ^ 4, 8x, 0, 1<D
2
15
Area2 = Integrate@x ^ 4 - x ^ 2, 8x, 1, 2<D
58
15
Area = Area1 + Area2
4
H* Find the volume of solid of revolution of the region between y=
x^2 and y=x^4 from x=0 to x=1 about the x axisL
VolumeWashers = Pi * Integrate@Hx ^ 2L ^ 2 - Hx ^ 4L ^ 2, 8x, 0, 1<D
4Π
45
VolumeShells = 2 * Pi * Integrate@y * Hy ^ H1  4L - y ^ H1  2LL, 8y, 0, 1<D
4Π
45
H* A tank has the shape of an inverted cone with base radius 5 feet and
height 15 feet. The vertex is below the base. It is filled with liquid
wieghing 50 lbfft^3. Find the work needed to pump all the liquid out of
the cone and up a spout which extends 1 foot above the top of the cone.*L
1
SolnEx1S14.nb
Work = Integrate@50  9 * Pi * y ^ 2 * H16 - yL, 8y, 0, 15<D H*units are ft-lb*L
59 375 Π
2
H*A spring force of 40 N is required to extend a spring 2 meters beyond
its natural length. Find the work required to stretch a spring from 10
meters beyond its natural length to 20 meters beyond its natural length*L
SpringConstant = 40  2
20
Work = Integrate@20 x, 8x, 10, 20<D H*units are Joules*L
3000
H* A 10 ft cable weighing 50 lb is hanging
vertically. Find the work done to pull up the cable *L
WorkCable = Integrate@5 * y, 8y, 0, 10<D
250
H*Bonus*L
H* Find the volume of the solid formed by revolving the region between y=
x^2 and y=x^4 from x=0 to x=1 about the x =1 axis*L
VolumeShells = 2 Pi Integrate@H1 - xL Hx ^ 2 - x ^ 4L, 8x, 0, 1<D
Π
10
y O - K1 2
VolumeWashers = Pi IntegrateBK1 -
y O , 8y, 0, 1<F
2
4
Π
10
H*Find the work done in pulling a bucket of water to the top a well that is 10 feet
deep. The bucket weighs 5 lb and is filled with 10 lb of water which leaks out
of a hole at a constant rate and 2 lbs of water are left after the bucket is
lifted 10 feet. The weight of the rope used to pull up the bucket is negligible.*L
WorkBucket = 5 * 10
50
WorkWater = Integrate@2 + 4  5 y, 8y, 0, 10<D
60
2
SolnEx1S14.nb
Work = WorkBucket + WorkWater
110
H*A tank is in the shape of a frustum of a cone. The smaller radius of
2 meters is on top and the larger radius of 10 meters is on bottom. The
vertical height is 8 meters. The tank is filled to a level of 2 meters frok
the top. Find the work done to pump out all the liquid from the tank. The
density of the liquid is 5 kgm^3. You may use 10 as an estimate of g *L
Solve@2  h Š 10  H8 + hL, hD
88h ® 2<<
H*Measure y from the imaginary vertex of the truncated cone*L
Work = 50 Pi Integrate@y ^ 2 Hy - 2L, 8y, 4, 10<D H*units are Joules*L
90 600 Π
3