Calculus II – Chapter 6 Practice Sheet
Name _________________________________
1. A spring has a natural length of 2 ft, and a force of 15 lb is required to hold it compressed at a length of 18 in. How
much work is done in stretching this spring from its natural length to a length of 3 ft?
2. A conical tank is upended so that its vertex is at ground level and its axis is vertical. The tank has radius 5 ft and its
height is 10 ft. Compute the work done in filling this tank with water pumped in from ground level.
3. Consider a spherical water tank whose radius is 10 ft and whose center is 50 ft above the ground. How much work is
required to fill this tank by pumping water up from ground level? (Suggestion: It may simplify your computations to
take y=0 at the center of the tank and to think of the distance each horizontal slice of water must be lifted.)
4. A tank in the shape of a hemisphere of radius 60 is resting on its flat base with the curved surface on top. It is filled
with alcohol of density 40 lb/ft3. How much work is done in pumping all the alcohol to the level of the top of the tank?
y  x  x 2 , x  0 , and x  1,
5. Find the surface area of revolution generated by rotating the region bounded by
around the x-axis.
6. Find the length of the smooth arc from x = 0 to x = 2 on the curve y 


3/ 2
2 2
x 1 .
3
7. Find the volume of the solid generated by rotating around the indicated axis the region bounded by the given curves.
a.
x  y, x  2 y  3,
y  0; x  axis
b.
y  x3 ,
c.
y  e x ,
d.
y  ex ,
y  0, x  0, x  1; x  axis
e.
y  x2 ,
y  8  x 2 ; the line y = -1
f.
y  x2 ,
y  4 x; the line x = 5
y  0, x  2;
2
about the line x = 3
y  0, x  0, x  1; y  axis
8. Find the area of the region bounded between the given curves.
a.
y  x2 ,
b.
y
1
x,
2
y  2x
y2  8  x