Find the volume of the region of space (or solid) above the graph of the function g(x,y) and below the graph of the function f (x,y) with the bounds on x and y as given below.  Here is a DPGraph picture of the solid (or region of space).

Click here or on the picture to see a Maple worksheet, including pictures, investigating this example including approximations to the volume using a sum of the volumes of approximating rectangular boxes.  The region along with one approximating rectangular box is pictured at the right.

Click here for an animation showing different views and the heights of each approximating box.  QT version

Click here or on the picture on the right to see an animation showing the region being filled with 10 approximating boxes.  Quicktime version    Click here to see the Maple worksheet for this example which includes constructing the pictures for the animation.  PowerPoint Show  

Click here to see a Maple worksheet for this example which includes the construction of pictures used in this animation which shows the volume approximated by 15 rectangular boxes.  Quicktime version    The picture at the right shows the region of the xy-coordinate plane partitioned into 15 equal sized rectangles with the height of each approximating rectangular box shown in red.

Click to see a Maple Worksheet relating to finding the volume of the solid bounded by the graphs of the surfaces described below and pictured at the right.

Find the volume of the solid in the first octant bounded by the graphs of

The volume would be computed by integrating z = 4 - y² over the region shown on the right.

Click on the picture above to see a picture of the solid.

Find the volume of the solid in the first octant bounded by the graphs of

Click on the picture above to see a picture of the solid.

DPGraphPicture    DPGraphPicture2

The volume is found by integrating the function 

over the region shown above.  (see DPGraph Top View)

Click on the picture above to see a picture of the solid.

DPGraph Picture     Maple Picture

Find the volume of the wall bounded by the graphs of the surfaces given below.  The region over which we would be iterating the double integral is pictured at the right.  Click on DPGraphPicture to see a 3D representation of the wall.  Click on DPGraphPicture2 to see the wall placed in an xyz-coordinate system.

Click on the picture above to see a Maple picture of the wall.

Find the volume of the region of space bounded by the graphs of z = x²+2y², z = 2, and z = 8.  See the graph on the right.  Click on the picture to see an animation.  DPGraphPicture     Maple Worksheet

Method 1

One option would be to integrate over the small ellipse, the level curve for z=2, with z going from 2 to 8 (i.e., the area of the small ellipse times 6) and add to this the result of integrating over the region between the two ellipses (between the level curve for z=2 and the level curve for z=8 which I am calling region R) with z going from x² + 2y² to 8.

An alternative would be to find the volume of the whole paraboloid up to z=8 and then subtract from this the volume up to z=2.

Method 3

Another option is to find the cross section area of a slice perpendicular to the z-axis and integrate this cross section area function of z from 2 to 8.

DPGraphPicture      DPGraphPic3Surfaces

The picture at the right shows the level curves for the first two surfaces described above corresponding to z = 5, cylinder in green, paraboloid in red.