Solids of Integration
A standard problem in a calculus class is find the volume of a solid
defined with a geometric base where the cross sections all have the
same shape. This applet is designed to help the student visualize
To define such a solid we need to define the base and the cross
section. Defining the base requires that we specify an xRange and
top and bottom curves with y as a function of x.
The default problem defines a base that is a circle of radius 5.
For that circle the xRange goes from -5 to 5, given as an ordered pair
in the control panel.
The default problem has a cross section that is a quarter
circle. The cross section is defined as a function of s with
range 0 to 1, named CrossSection in the control panel.
The problem assumes that the student will use geometry to find the area
cross section. The user is expected to provide the
CrossSectionArea for the shape when the base is of length 1. The
area of an arbitrary cross section is the area of this cross section
times the square of the base of the cross section.
The applet is launched by clicking on the button. You will
see a control window and two graphical windows. The 2-D
graphical window shows the base in green and a cross section in
red. The 3-D window shows the volume of integration. The
solid can be rotated by dragging. You can zoom in or out with the
The size of the graphical windows can be changed with the usual window
control in the lower left corner of the windows. The check box in the
control window lets you remove the top to see just
the base and the cross section with x=x0. The read out boxes give
the area of that cross section and the volume of the solid.
The table gives the functions for a variety of cross sections.
|Isosceles right triangle
This applet was designed as by modifying an applet from a demo by
Tom Banchoff at
Brown University. It is used
Go to the Banchoff
Help page for more information on using the applet, or for a list of
Return to the Banchoff Applet page.
Return to the SLU Calculus
Return to the Saint Louis University Department of Mathematics and Computer Science
Last updated By Mike May, S.J.
May 18, 2007.