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 such solids.

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 of a 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 view menu.  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.
 Shape Function Section Area Square 1 1 Isosceles right triangle s 1/2 Quarter circle sqrt(1-s^2) Pi/4 Semi-circle sqrt(1-(2*s-1)^2)/2 Pi/8

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.
Go to the Banchoff Applet Help page for more information on using the applet, or for a list of available functions.