7.2 Finding Volumes by Integration
Volumes of Revolution: Disk Method
This applet is for use when finding volumes of revolution using the disk method when rotating an area between a function f(x) and either the x- or y-axis around that axis. As usual, enter in the function of your choice. Select (and/or de-select) the appropriate axis of revolution. Set upper and lower bounds on the region. Use the slider to rotate the region around the appropriate axis.
Volumes of Revolution: Washer Method
This applet is for use when finding volumes of revolution using the washer method when rotating an area between two functions f(x) & g(x) around a line. To start, select either "rotate around y=?" or "rotate around x=?" and then enter the appropriate line. Next, enter the appropriate choices for the inner and outer functions. Set upper and lower bounds on the region. Use the slider to rotate the region around the appropriate axis and observe the construction of the solid.
Solids with Known Cross Sections
This applet shows a graphical view of a solid with cross sections perpendicular to the xy-plane while the base is given by a region enclosed in the xy-plane. This applet is only suitable for use when the base of the region can be described by two curves that bound the top and bottom of the region, the curves are entered as f & g, respectively. Select the appropriate cross-sectional area: square, semi-circle, equilateral triangle, or isosceles triangle. To avoid confusion, select only one cross-section at a time. After entering lower and upper bounds for the x-values, you can adjust the slider named "xval" to trace out the solid from the lower to upper endpoints of the interval of interest. Note: you may need to alter the "increment" size to give you a better display. As usual, you can move the cartesian coordinate plane with your mouse and zoom in and out by scrolling. Example: Suppose the base of a solid object is the circle ; cross sections of the object perpendicular to the x-axis are squares with one side in the xy-plane. Define the top curve as (found by solving for ) and the bottom curve as (the bottom of the region is bounded by the x-axis). Select "square" as the only cross section. Because the graph of intersects the x-axis at , we set the upper bound as 2 and the lower bound as -2. Our interval is 4 units wide, so we want our increment to be smaller than the interval width, for this example we will pick the increment value to be 0.5 . As you move the slider from to , you will see a frame-like cross section every 0.5 units while the overall solid is shaded in.