When the figure first loads, we see a region bounded below by [math]y=0.1x^2[/math], bounded above by [math]y=4-\frac{x^2}{8}[/math], and bounded on the sides by the lines [math]x=1[/math] and [math]x=3[/math]. A function [math]f\left(x,y\right)[/math] is also given and we wish to integrate [math]f\left(x,y\right)[/math] over this region.[br] Click the checkbox next to [math]f\left(x,y\right)[/math] to show/hide the surface [math]z=f\left(x,y\right)[/math]. You can drag the 3D view to get a better perspective. Clicking the button "3D [math]\rightarrow[/math] 2D" will also change the view. Computing the double integral tells us the volume of the solid above the bounded region and below [math]z=f\left(x,y\right)[/math].[br] First, we compute [math]A\left(x\right)=\int_{0.1x^2}^{4-x^2\slash8}f\left(x,y\right)dy[/math]. Thus, [math]A\left(x\right)[/math] tells us the cross-sectional area of the solid for any [math]x[/math]. Next, we integrate [math]A\left(x\right)[/math] from [math]x=1[/math] to [math]x=3[/math] to get the volume of the iterated integral.

If you make your own examples, you may need to change the 3D view to see your bounded region and the solid clearly. [br][list][*]Dragging rotates the view. [/*][*]Shift+Drag will pan the view. [/*][*]Shift+Click will change the direction of the panning. [/*][*]Scroll wheel zooms in and out.[br][/*][/list][i]Developed for use with Thomas' Calculus, published by Pearson.[/i]