1
Name:
Class: AP Calculus
Per:
Date: Thursday March 31st, 2011
Topic: Volumes of solids with known cross sections
Aim: How can we find the volumes of solids with known cross sections?
Objectives:
1) Students will be able to find the volumes of solids with known
cross sections.
HW# 78: Find
the volume of the solid that lies between
planes perpendicular to the y-axis at y = 0 and y = 2.
The cross sections perpendicular to the y-axis are
circular disks with diameters running from the y-axis to
the parabola.
Do Now: At right you have a circle with equation x 2  y 2  9 .
Express the length of the chord in terms of y
Answer:
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
If the area of a cross section of a solid is known and can be expressed in terms of y or x , then the volume of a typical
slice can be determined. The volume of the solid is obtained, as usual, by letting the number of slices increase
indefinitely.
2
You can use the definite integral to find the volume of a solid with specific cross sections on an interval.
If the cross sections generated are perpendicular to the x-axis, then their areas will be functions of x, denoted by A(x).
b
V   A( x)
The volume ( V) of the solid on the interval [ a, b] is
a
If the cross sections generated are perpendicular to the y-axis, then their areas will be functions of y, denoted by A(y).
b
V   A( y )
The volume ( V) of the solid on the interval [ a, b] is
a
Example 1: Find the volume of the solid whose base is the region inside the
circle x2 + y2 = 9 if cross sections taken perpendicular to the y-axis are squares.
Because the cross sections are squares perpendicular to the y-axis, the area of
each cross section should be expressed as a function of y. Solving for x gives
us x  9  y 2 . That is half the length of the bottom of the square. The
entire bottom has length 2 9  y 2 . Then the area of the square becomes

 4(9  y )dy

3
4 9  y 2 . Then the volume of the solid becomes
2
3
Answer:
Example 2: Find the volume of the solid whose base
is the region bounded by the lines x + 4 y= 4, x = 0,
and y = 0, if the cross sections taken perpendicular to
the x-axis are semicircles.
Because the cross sections are semicircles
perpendicular to the x-axis, the area of each cross
section should be expressed as a function of x
1 2
r . The radius is
2
half the length between the line x  4 y  4 and
The area of the semi circle is
y  0 . Solve for y in the first one and we get y 
4 x
4 x
. So the radius is half of that, namely
4
8
1 4 x
So the area of each cross section is  
 . So the volume of the solid is
2  8 
2
Answer:
1 4 x
0 2   8 
4
2
3
On Your Own:
1) Find the volume of the solid that lies between planes perpendicular to the x-axis at x = 0 and x
= 4. The cross sections perpendicular to the x-axis between these planes run from one side of the
parabola x = y 2 to the other. The cross sections are squares with bases in the xy-plane.
2) Find the volume of the solid that lies between the planes perpendicular to the x-axis at x = -1
and x = 1. The cross sections perpendicular to the x-axis are circular disks whose diameters run
from the parabola y = x 2 to the parabola y = 2 - x 2.
4
3) A solid has its base is the region bounded by the lines
x + 2y = 6, x = 0 and y = 0 and the cross sections taken
perpendicular to x-axis are circles. Find the volume the
solid.
4) A solid has its base is the region bounded by the lines x + y = 4,
x = 0 and y = 0 and the cross section is perpendicular to the x-axis
are equilateral triangles. Find its volume.