1.) Draw the solid S.
2.) Find equations for each of the smooth pieces of the boundary of S.
3.) Decide which variable you want to use for the first integration. This is similar to the two variable case. In your mind draw a set of lines parallel to the z axis, another set parallel to the y axis, and another set parallel to the x axis. Look at where these lines enter and leave the solid. If possible choose a set in which all the lines enter by one piece of boundary, and leave by another. We'll suppose we picked a set parallel to the z axis. This means we should integrate with respect to z first.
4.) Look at where the vertical segments enter the solid S. Use the equations you found in step 2 to write a formula for the z value where the segments first enter the solid in terms of x and y. This is the lower limit of the z integral.
5.) Look at where the vertical segments leave the solid S. Use the equations you found in step 2 to write a formula for the z value where the segments leave the solid in terms of x and y. This is the upper limit of the z integral.
6.) Draw the shadow of the solid. The shadow is the set of points in xy-plane which would be shaded if we put a light over our shape. In more precise language, each vertical line pases through some point on the xy-plane. That point is in the shadow, if its vertical line intersects the solid.
7.) Find the limits of integration for the integral over the shadow. This you know how to do, because the shadow is a region in the plane. (If you forget go to How to find the limits of integration, given a region.)
To show you how the procedure works, we're going to use the following problem:
Set up the integral of f(x,y,z) over the solid with upper boundary the surface with equation z= 25-x2-y2 and lower boundary the plane with equation z= 9.
1.) Draw the solid S.
In our example, the top of the solid is an upside down bowl with vertex at (0,0,25). The bottom of the solid is the plane with equation z=5. ( For help in drawing the bowl go to Problem 1 on Quiz 1.) When we draw the solid we get:
2.) Find equations for each of the smooth pieces of the boundary of S.
These are given as part of the problem.
3.) Decide which variable you want to use for the first integration.
Since all of the vertical lines enter by the lower bounary and exit by the upper boundary, we can integrate with respect to z first without splitting up the integral.
4.) Look at where the vertical segments enter the solid S. Use the equations you found in step 2 to write a formula for the z value where the segments first enter the solid in terms of x and y. This is the lower limit of the z integral.
Since the equation of the lower bounary is z=5, 5 is the lower limit for the first integral.
What is the upper limit?
Click here for the next step.
Return to Quiz 9.