Summary of area formula The area of the region bounded by and

is

The area of the region bounded by and

is

( ), ( )y f x y g x , ( )x a x b a b

| ( ) ( ) | .b

aA f x g x dx

| ( ) ( ) | .d

cA f y g y dy

( ), ( )x f y x g y , ( )y c y d c d

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

2 2 ( ) .b b b

a a aV dV y dx f x dx

2 2 ( ) .d d d

c c cV dV x dy f y dy

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

2 2 ( ) .b b b

a a aV dV xydx xf x dx

2 2 ( ) .d d d

c c cV dV yxdy yf y dy

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.

2 2

0 0(1) (1 sin ) or sin (2 sin )x dx x x dx

0, sin , 0 ;y y x x 1y

3 / 2 2

0(3) sin .xdx

0, sin , 0 ;y y x x / 4x 0, sin , 0 3 / 2;y y x x

/ 4(2)2 ( )sin

4x xdx

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

( ) ,W f x dx dW

( ) .b b

a aW dW f x dx

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/(0.15-0.1)=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 is0.08

0.05800 1.56.W xdx

Example Ex. A container which has the shape of a half ball with rad

ius 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

[0, ],x R

2 2( ) .V R x dx dV 2 2( ) .dW dV gx g x R x dx

2 2 4

0

1( ) .

4

RW g x R x dx R g

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

1( ) .

b

ave af f x dx

b a

( ) ( )( ).b

af x dx f c b a

Homework 16 Section 6.2: 14, 18

Section 6.3: 7, 14

Page 470: 3