Objectives
After completing this section you will:
Be comfortable setting up triple integrals for a solid region (domain)
Be able to switch the order of integration for a triple integral
After completing this section you will:
Be comfortable setting up triple integrals for a solid region (domain)
Be able to switch the order of integration for a triple integral
Consider the iterated integral
∫3−3∫√9−y20∫9−x2−y201dzdxdy.∫3−3∫√9−y20∫9−x2−y201dzdxdy.This is an integral of the form ∭DdV,∭DdV, which means along some solid region DD in the plane, we are adding up little bits of volume. This integral will give the volume of a very familiar solid region in space. Not every triple integral will correspond to an easily drawn or visualized (even with a computer) space. However, this one does.
Before we compute this integral, let's visualize it. The parenthesis have been added to help you parse this first example.
∫3−3(∫√9−y20(∫9−x2−y201dz)dx)dy.∫3−3(∫√9−y20(∫9−x2−y201dz)dx)dy.State the bounds for each variable as inequalities.
Sketch the region DD in space.
Compute the innermost integral, and compare the resulting/remaining double integral to the first exercise in the double integral unit.
Now evaluate the remaining integrals (though you might want to change coordinate systems before doing so).
When working with double integrals, there were two different ways to set up the bounds for our integrals, as dA=dxdy=dydx.dA=dxdy=dydx. When working with triple integrals, there are six different ways to order the differentials, and therefore set up bounds for our integrals.
dV=dxdydz=dxdzdy=dydxdz=dydzdx=dzdxdy=dzdydxdV=dxdydz=dxdzdy=dydxdz=dydzdx=dzdxdy=dzdydxConsider again the iterated integral
∫3−3∫√9−y20∫9−x2−y20dzdxdy=∫90∫√9−z0∫√9−x2−z−√9−x2−zdydxdz∫3−3∫√9−y20∫9−x2−y20dzdxdy=∫90∫√9−z0∫√9−x2−z−√9−x2−zdydxdzThese two integrals are equal, but required switching the order of the bounds. There are 4 other iterated integrals that are equal to these integral which can also be found by switching the order of the bounds.
Set up the equivalent integrals using the differential orderings: dydzdxdydzdx and dxdzdy.dxdzdy. We'll look at the remaining 2 in class (though you're welcome to finish them and present them with your work).
Consider the iterated integral
∫1−1∫1−x20∫y0dzdydx.∫1−1∫1−x20∫y0dzdydx.The bounds for this integral describe a region in space which satisfies the 3 inequalities −1≤x≤1,−1≤x≤1, 0≤y≤1−x2,0≤y≤1−x2, and 0≤z≤y.0≤z≤y.
Draw the solid domain DD in space described by the bounds of the iterated integral.
There are 5 other iterated integrals equivalent to this one.
Set up the integral that use the differential ordering: dydxdzdydxdz
Set up the integral that use the differential ordering: dxdzdy.dxdzdy.
We'll create the other 3 in class (though you are welcome to include them as part of your presentation)
In each exercise below, you'll be given enough information to determine a solid domain DD in space.
Draw the solid DD and
Set up an iterated integral (pick any order you want) that would give the volume of D.D.
Note: You don't need to evaluate the integral, rather you just need to set them up.
The region DD under the surface z=y2,z=y2, above the xyxy-plane, and bounded by the planes y=−1,y=−1, y=1,y=1, x=0,x=0, and x=4.x=4.
The region DD in the first octant that is bounded by the coordinate planes, the plane y+z=2,y+z=2, and the surface x=4−y2.x=4−y2.
The pyramid DD in the first octant that is below the planes x3+z2=1x3+z2=1 and y5+z2=1.y5+z2=1.
Don't let \(z\) be the inside bound.
The region DD that is inside both right circular cylinders x2+z2=1x2+z2=1 and y2+z2=1.y2+z2=1.
We can find average value, centroids, centers of mass, moments of inertia, and radii of gyration exactly as before. We just now need to integrate using three integrals, and replace ds,ds, dAdA or dσ,dσ, with dV.dV.
Consider the triangular wedge DD that is in the first octant, bounded by the planes y7+z5=1y7+z5=1 and x=12.x=12. In the yzyz plane, the wedge forms a triangle that passes through the points (0,0,0),(0,0,0), (0,7,0),(0,7,0), and (0,0,5).(0,0,5).
Set up integral formulas that would give the centroid (ˉx,ˉy,ˉz)(¯x,¯y,¯z) of D.D.
It may help to sketch a picture first
Actually compute the integrals for ˉy.¯y.
Now state ˉx¯x and ˉz¯z by using symmetry arguments.
These are provided to help you achieve better skills in basic computational answers.