Part II
1. (Area of a plane region)
1. Region bounded by x-axis
2. Region between two curves
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(a) Find the area of the region bounded by f(x) = x3
- 4x and the x-axis.
(b) Find the area of the region bounded by y = x2
- 6x and y = -x2 + 6x.
2. (Volume of a solid of revolution)
Sketch the region R bounded by the parabola
, the line x = 1, and the x-axis
A solid is generated by revolving the region R about the
line y = -1. Set up (but do
not evaluate) definite integrals for the volume of the resulting solid of
revolution using
(a) washers,
(b) shells
3. (Length of a plane curve, Area of a surface of
revolution)
1. Length of curves
where
2. Area of a surface of revolution
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(a) Find the length of the curve x = cos 2t, y = sin 2t,
and 0 ≤ t ≤ 2π .
(b) Find the area of the surface generated by revolving
the curve 
, 1 ≤
x ≤ 2 about the x-axis.
4. (Work d one by a variable force)
Pumping the water
First, place a vertical cross section on the coordinate system .
where
a = the bottom of the water
b = the top of the water
 = density of water
A(y) = area of horizontal cross section at y
d(y) = distance to which the cross section will be lifted
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(a) A swimming pool has a rectangular base 10 ft long and
12 ft wide. The sides are 6
ft high, and the pool is half full of water. How much work will it take to empty
the
pool by pumping the water out over the top of the pool?(Set up only.)
(b) A right circular conical tank of base radius 4 feet
and height 8 feet is initially full
of a liquid with density
, find the work done in pumping all
liquid to a height 2
feet above the top of the tank. (Set up only.)
5. (Moments and center of mass of a plane lamina,
centroid of a plane region.)
Moments and Centroid of a plane region
Total moments: ( : with respect to
y-axis, and
 : with respect to x-axis)
Total mass: 
Centroid: 
Pappus's Theorem
V = 2 πdA,
where d is the distance between the revolving axis and the centroid, and
A is the area of the region.
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(a) Find the centroid of the region R between the curve y
= cos x and the x-axis
between x = - π/2 and x = π/2.
(b) Find the volume of the solid obtained when R in (a) is
revolved around y =
-1.(Pappus's Theorem)
