Calculus Section 7.2
Volumes by Slicing and Disks and Washers
Homework:
Day 1
Day 2
Page 456 #1 – 25 odd and #29
Page 457 # 35 – 43 all (skip 39, 40)
Objective: SWBAT find the volume of three-dimensional objects using the definite
integral.
1. Volume of a right cylinder
2. Volume formula (slicing or known cross sections)
a. Let S be a solid bounded by two parallel planes perpendicular to the x-axis at
x = a and x = b. If, for each x in [a,b], the cross-sectional area of S perpendicular
to the x-axis is A(x), then the volume of the solid is
b
V   A( x)dx
a
provided A(x) is integrable
b. Let S be a solid bounded by two parallel planes perpendicular to the y-axis at
y = c and y = d. If, for each y in [c,d] , the cross-sectional area of S
perpendicular to the y-axis is A(y), then the volume of the solid is
d
V   A( y)dy
c
d
V   A( y )dy
c
provided A(y) is integrable
3. Slicing rules (known cross sections)
a. Sketch the base of the solid in the regular xy-coordinate plane.
b. Sketch or envision the solid. Usually the x-axis is drawn coming “out
of the paper” toward you to give the 3D effect.
c. Determine whether the integral should be in terms of x or y.
If the cross section is perpendicular to:
x- axis = dx
y – axis = dy
d. Determine the bounds of integration by finding the extent of the base
along the x or y – axis.
e. Use geometry and the given equation(s) to find an expression for the
area of the cross section.
f. Set up the integral and integrate.
4. Example (slicing or known cross sections)
Find the volume of the solid whose base is a circle x2 + y2 = 9 and
whose cross sections are perpendicular to the x-axis are squares. Let us
look at the base of some of the squares.
7
6
5
4
3
3
y =  9-x2
2
1
-6
-4
-3
-2
2
3
4
-1
-2
-3
-3
-4
Now let us rotate the plane so we can see the squares.
6
8
7
6
5
4
3
3
y =  9-x2
2
1
-6
-4
-3
-2
3
2
4
6
-1
-2
-3
-3
Now let us find the area of the shape created below.
8
7
6
5
4
3
3
y =  9-x2
2
1
-6
-4
-2
2
-1
-2
3
4
6
8
10
8
5. Example
Find the volume of the solid of revolution whose base is bounded by the lines
f ( x)  1  x
g ( x)  x  1
x0
and whose cross sections are semicircles perpendicular to the x-axis.
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
P
0.4
0.2
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
-2
-2.2
-2.4
Q
1.5
2
2.5
3
3.5
6. Volume by disks or washers
a. Sketch the original area.
b. Sketch the solid of revolution.
c. Determine if you should disks or washers
*d. If the axis of revolution is horizontal integrate with respect to x (dx)
*e. If the axis of revolution is vertical integrate with respect to y (dy)
f. Identify the parts of the integrand, sketch and label as necessary. If needed,
substitute from the original equation so that all parts are in terms of x or y only as
determined by d and e above.
g. Determine the bounds of integration by examining the extent of the original
area along the x-axis or the y-axis to correspond to d and e above.
Set up the integral, don’t forget the  .
h. Integrate and substitute.
7. Volume using DISKS
Diagram
8. Formula
b
V    [ f ( x)]2 dx
a
d
V    [ f ( y)]2 dy
c
9.
volume when revolving around the x-axis
volume when revolving around the y-axis
Example using DISKS
Find the volume of the solid that is obtained when the region under the curve
y  x over the interval [1,4] is revolved about the x-axis.
10. Example (Hershey Kiss example)
Use the disc method to find the volume of the solid of revolution generated when
an area is bounded by
x  axis
x2
y  x2
and revolved about the line x  2 .
11. Method using WASHERS
Diagram
12. Formula
b
V    ([ f ( x)]2  [ g ( x)]2 )dx
a
d
V    ([ f ( y)]2  [ g ( y)]2 )dy
c
13. Example using Washers
Find the volume of the solid generated when the region between the graphs of
1
 x2
2
g ( x)  x
f ( x) 
over the interval [0,2] is revolved about the x-axis.
14. Example Washer Method (Cup)
Use the washer method to find the volume of the solid of revolution generated when
an area is bounded by
x  axis
x2
y  x2
and revolved about the y-axis.
15. Example
a.
Find the volume of the solid generated when the region bounded by y = x2
and y = 4x – x2 is revolved around the x-axis.
Extra Practice 7.2
(Watts page 392 # 35,36)
1. Find the volume of the solid generated by revolving the region bounded by y  x
x  0 about:
and the lines y  2 and
a. The x-axis
b. The y-axis
c. The line y  2
d. The line x  4
2. Find the volume of the solid generated by revolving the region bounded by the
parabola y  2 x and the line y  0 and x  1 about the line x  1
3. Find the area enclosed by the region bounded by y  tan( x), y  1, x  0 and the xaxis. Integrate with respect to x and with respect to y.
4. (Summer Institute 4A) Find the volume of the solid whose base is bounded by
y  9  x2
y  ex
x0
and whose cross sections cut by planes perpendicular to the x-axis are squares.
(Summer Institute 32B,32C)
5. Find the volume of the solid whose base is enclosed by the graph of y  x , y  0 ,
and x  4 whose cross sections cut by planes perpendicular to the x-axis are:
a. Squares
b. Rectangles with heights of 2
c. Semi Circles
6. Find the volume of the solid whose base is enclosed by the graph of y  x 2  1 , y  0 ,
x  0 , and x  1 whose cross sections cut by planes perpendicular to the x-axis are:
d. Squares
e. Rectangles with heights of 2
f. Semi Circles