1. Summary of area formula
The area of the region bounded by and is

2. Summary of volume formula
The volume of the solid obtained by rotating about x-axis the region enclosed by y=f(x), x=a, x=b and x-axis, is
The volume of the solid obtained by rotating about y-axis the region enclosed by x=f(y), y=c, y=d and y-axis, is

3. Volume by cylindrical shells
The volume of the solid obtained by rotating about y-axis the region enclosed by y=f(x), x=a, x=b and x-axis, is
The volume of the solid obtained by rotating about x-axis the region enclosed by x=f(y), y=c, y=d and y-axis, is

4. Example Ex. Set up an integral for the volume of the solid
obtained by rotating the region bounded by the given curves
about the specified line:
(1) about
(2) about
(3) about x-axis
Sol.

5. Physical application: work
Problem: Suppose a force f(x) acts on an object so that it
moves from a to b along the x-axis. Find the work done by
the force f(x).
Solution: take any element [x,x+dx], the work done in
moving the object from x to x+dx is
so the total work done is

6. Example Ex. A force of 40N is required to hold a spring that has
been stretched from its natural length of 10cm to 15cm. How
much work is done in stretching the spring 3cm further?
Sol. By Hooke’s Law, the spring constant is
k=40/( )=800.
Thus to stretch the spring from the natural length 0.1 to 0.1+x,
the force will be f(x)=800x. So the work done in stretching it
from 0.15 to 0.18 is

7. Example
Ex. A container which has the shape of a half ball with radius R, is full of water. How much work required to empty the container by pumping out all of the water?
Sol. We first set up a coordinate system: origin is the
center of the ball and vertical downward line is x-axis.
For any take an infinitesimal element [x,x+dx].
The water corresponding to this small part has volume
To pump out this part of water, the
work required is
Therefore total work is

8. Average value of a function
The average value of f on the interval [a,b] is defined by
The Mean Value Theorem for Integrals If f is
continuous on [a,b], then there exists a number c such that

9. Homework 16
Section 6.2: 14, 18
Section 6.3: 7, 14
Page 470: 3