Triple Integrals: Setting up the Integral

Transcription

1 Triple Integrals: Setting up the Integral. Set up the integral of a function f x, y, z over the region above the upper nappe of the cone z x y from z to z. Use the following orders of integration: d x d y d z, d y d x d z and d z d x d y. Solution: The region of integration is ContourPlotDz^ x^ y^, z, z, x,,, y,,, z,,., ContourStyle DirectiveOpacity., DirectiveOpacity.8, DirectiveOpacity.8 For the d x d y d z order of integration we have

2 Triple_Integral_Examples.nb To find the y and z limits, we look at the projection of our region onto the yz - plane: ContourPlotz^ y^, y,,, z,, For a fixed z, we see that y goes from the line on the left to the line on the right. The left line has equation z y and the line on the right has equation z y. This explains the limits on the second integral. The limits on the z integral are clearly and. For the d y d x d z order of integration we have z x x z x above only with the horizontal axis being the x - axis. Hence the integral is. For the x limits, we have the same picture as that just Finally, for the d z d x d y order of integration, we have a more complicated situation. The integral breaks into two parts:. The portion of the region inside the cone that lies above the circle x y. Here, z runs from to.. The portion outside the circle x y and inside the circle x y. Here, z runs from the cone up to z. For (), the integral is

3 Solution: A plot of this region is shown below: Triple_Integral_Examples.nb since we are directly above the circle of radius centered at the origin and z runs from to over this circle. For () we have, for limits on z, x y z. The region in the xy-plane is the annular region shown below: ContourPlotx^ y^, x,,, y,,, RegionFunction Functionx, y, z, x^ y^ 8, BoundaryStyle Red We can use symmetry to reduce the problem here to the first quadrant and multiply by. For any fixed y such that y, x runs from the y-axis to the outer circle: x 8 y. For any fixed y such that y, x runs from the inner circle to the outer circle: y x 8 y. Hence our integral for () is: 8y y x y f x, y, z z x y 8y x y f x, y, z z x y The final integral for this order of integration is y y f x, y, z z x y 8y y x y f x, y, z z x y 8y x y f x, y, z z x y. Set up the integral of a function f x, y, z over the region bounded above by z 8 x y and below by z x y using the best choice of the order of integration. Find the volume of this region using a triple integral.

4 Triple_Integral_Examples.nb. Set up the integral of a function f x, y, z over the region bounded above by z 8 x y and below by z x y using the best choice of the order of integration. Find the volume of this region using a triple integral. Solution: A plot of this region is shown below: ContourPlotD8 x^ y^ z, z x^ y^, x,,, y,,, z,., 8., Mesh None, ContourStyle Opacity., Opacity. Looking at this plot, we see that the easiest order of integration is d z d y d x (or d z d x d y). The region in the xy-plane that we integrate over is found by setting 8 x y x y. We get 9 x y - an ellipse. This ellipse is shown below:

5 Triple_Integral_Examples.nb ContourPlot9 x^ y^, x,,, y,, Clearly the region in the xy - plane is both x and y simple. Our integral is: The volume of this region is simply the integral above with f x, y, z. We have: 9 x 9 x 8x y x y z y x 7 Π. Find the volume of the region bounded by the graphs of z x, z x, y, and z y. Solution: As seen below, the region lies under the cylinder z x, over the cylinder z x, to the right of the xz-plane and to the left of the plane z y. Hence the order of integration to use here is d y d z d x or d y d x d z.

6 Triple_Integral_Examples.nb ContourPlotDz x^, z x^, z y, y, x,.,., y,,., z,,., ContourStyle Opacity., Opacity., Opacity.9, Opacity.9, Mesh None The region in the xz-plane to integrate over is Plot x^, x^, x,,, AxesLabel x, z, Filling z z x z x.... x This plot clearly shows us that we will need to use a d z d x ordering for the outer two integrals. The integral for the volume is therefore

7 Triple_Integral_Examples.nb The integral evaluates to: x z x y z x