### Section-7.3cx

```DO NOW: Find the volume of the solid generated when the
region in the first quadrant bounded by the given curve and line
is revolved about the x-axis. y  2 x4  3x2  5
x2
(2,25)
Cross-section area:
A x    r    2 x  3x  5 
8
6
4
2
   4 x  12 x  29 x  30 x  25 
2
(0,5)
4
2
2
Volume:
f(x)
x
V     4 x8  12 x6  29 x 4  30 x 2  25 dx
2
0
2
 4 9 12 7 29 5

3
   x  x  x  10 x  25x 
7
5
9
0
51574

315
The Washer Method
SECTION 7.3C
The region in the first quadrant enclosed by the y-axis and the
graphs of y = cos(x) and y = sin(x) is revolved about the x-axis
to form a solid. Find its volume.
Graph the region… and visualize the solid…
 0,1 
4, 2 2

Each cross section perpendicular to the
axis of revolution is a washer, a circular
region with a circular region cut from
its center:
R
r
Area of a washer:
 R  r
2
2
The region in the first quadrant enclosed by the y-axis and the
graphs of y = cos(x) and y = sin(x) is revolved about the x-axis
to form a solid. Find its volume.
 0,1 
4, 2 2

The outer and inner radii are the y
values of our two functions!!!
R  cos x
r  sin x
Cross section area:
A  x     cos x  sin x 
2
Volume:
V 
 4
0

 4
0
2
  cos2 x  sin 2 x  dx
 4
1
 
cos 2xdx    sin 2 x 
2
2
0
units
cubed
Guided Practice
Find the volume of the solid generated by revolving the region
bounded by the given lines and curves about the x-axis.
1, 2
1,1
y  2x y  x x  1
Cross section area:
A x  
 2x  x   3 x
2
Volume:
1
V   3 x dx
2
0
1
x 
 3    
 3 0
3
2
2
Guided Practice
Find the volume of the solid generated by revolving the region
bounded by the given lines and curves about the x-axis.
y  4 x
 1,3
2
y  2 x
Cross section area:
2

A  x     4  x    2  x  


 2, 0 
2
4
  12  4 x  9 x  x 
2 2
Volume:
V    12  4 x  9 x  x  dx
2
2
4
1
2

x  108
2
3
  12 x  2 x  3 x   
5  1
5

5
Guided Practice
Find the volume of the solid generated by revolving the given
The region bounded above by the curve y  x and below
by the line y  x .
Cross section area:
x  y2
 
 y  y 

A
y


y

y


1,1
 
x y
2
2
2 2
4
y
 2
y
Volume: V    y  y  dy  
 3  5   15
0

0
1
3
2
4
5
1
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region in
the first quadrant bounded above by the line y  2 , below by the
curve y  2sin x , 0  x   2 , and on the left by the y-axis,
about the line y  2 .
r  2  2sin x
Cross section area:
A x    r    2  2sin x 
2
r

2, 2
 4 1  sin x 
2
2
 4 1  2sin x  sin 2 x 
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region in
the first quadrant bounded above by the line y  2 , below by the
curve y  2sin x , 0  x   2 , and on the left by the y-axis,
about the line y  2 .
Volume:
V 
 2
0
r

2, 2
 4 
4 1  2sin x  sin x  dx
 2
0
2
1 1


1  2sin x   cos 2 x  dx
2 2


1
3

 4    2sin x  cos 2 x  dx
0
2
2

 2
1
3

 4  x  2 cos x  sin 2 x     3  8
4
2
0
 2
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the triangular
region bounded by the lines y = 2x, y = 0, and x = 1 about
(a) the line x = 1.
1, 2
1
Cross section radius: r  1 
y
2
Cross section area:
 1    1  y  1 y 2 
A  y    1  y 


4 

 2 
2
r
Volume:
1 2

V    1  y  y  dy
0
4 

2
2
1 2 1 3

 y  y  y   
2
12  0 3

2
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the triangular
region bounded by the lines y = 2x, y = 0, and x = 1 about
1
Washers!!! r  1 R  2 
y
2
Cross section area:
(b) the line x = 2.
x2
2


1 
2
A  y     2  y   1 
2 


1 2

  3 2y  y 
4 

r
Volume:
R


y 
y  8
2
V     3  2 y   dy   3 y  y   
0
12  0 3
4 


2
4
3
2
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
2
bounded by the parabola y  x and the line y  1 about
(a) the line y = 1.
Cross section:
r  1 x
A  x    1  x

2 2
2
  1  2x 2  x 4 
Volume:
V    1  2 x  x  dx
1
2
4
1
 2 3 1 5  16
  x  x  x  
5  1 15
 3
1
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
2
bounded by the parabola y  x and the line y  1 about
(b) the line y = 2.
Washers:
r  1 R  2  x2
2
4
2 2
2

A  x     2  x   1     3  4x  x 


Volume:
V     3  4 x  x  dx
1
2
4
1
4 3 1 5  56

  3 x  x  x  
3
5  1 15

1
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
2
bounded by the parabola y  x and the line y  1 about
(c) the line y = –1.
Washers:
r  1  x2 R  2
2
4
2
2 2

A  x    2  1  x      3  2x  x 


Volume:
V     3  2 x  x  dx
1
2
4
1
2 3 1 5  64

  3 x  x  x  
3
5  1 15

1
```