Evaluating Triple Integral of Region E

[stratusfactio]

Homework Statement

Evaluate the triple integral of the region E, where E is the solid w/i the cylinder x^2+y^2=1, above the plane z=0, and below the cone z^2=x^2+y^2.

So is the plane z=0 same as the xy-plane? I was doing a homework problem that has

Homework Equations

I just need help conceptually understand the limits of integration. So we know that x has intercepts at ±1 and y has intercepts at ±1. And since the cylinder lies along the z-axis, the radius integrand ranges from -1 to 1. The theta integrand ranges from 0 to 2∏. And the z integrand ranges from 0 to 2r.

Now when I saw the solution, it said that the radius integrand ranges from 0 to 1; not -1 to 1. Which makes me question, is the plane z=0 a vertical plane or horizontal plane? Or do you think the solution has an error? Because if the plane was vertical, the radius integrand would range from 0 to ∏

The Attempt at a Solution

 

[Simon Bridge]

Homework Statement

Evaluate the triple integral of the region E, where E is the solid w/i the cylinder x^2+y^2=1, above the plane z=0, and below the cone z^2=x^2+y^2.

So is the plane z=0 same as the xy-plane?

Yes.

Homework Equations

I just need help conceptually understand the limits of integration. So we know that x has intercepts at ±1 and y has intercepts at ±1. And since the cylinder lies along the z-axis, the radius integrand ranges from -1 to 1.

A radius cannot be negative.
Try it - draw a circle with a negative radius. You can't do it right?

The theta integrand ranges from 0 to 2∏. And the z integrand ranges from 0 to 2r.

Where did this r come from?
You know how high the cone is?

You realize that the volume you want is the cylinder minus the cone right?

To help you visualize what is going on - try:
1. sketching the cross-section in the y-z plane - should be an M shape.
2. work out the answer without doing the integration

 

[Coto]

First off, what type of coordinate system are you using? Cylindrical? Cartesian? This is among the first decisions you must make when solving an integral of this type.

It sounds like you are choosing a cylindrical coordinate system, in which case your coordinates are of the form (r,\theta,z).

For your first question then, radius is strictly a positive quantity (have you ever heard of a circle with a negative radius?) and it denotes the distance from the z-axis to a point (x,y) in the same horizontal plane (assuming z denotes the vertical axis).

If you've drawn the z-axis vertically, then the z=0 plane denotes a horizontal plane. However, this is strictly arbitrary and dependent upon how you draw your axes. For this reason, it doesn't make sense to denote the orientation of the plane with the words "horizontal" and "vertical". A more correct wording would be that the z=0 plane is the xy-plane, as you suspected.

This type of integral represents the volume of an object (in this case an inverted cone bounded by a cylinder), and as such, does not depend on the orientation at all (i.e. it doesn't matter from which direction you look at it, it'll always have the same volume.)

To derive the limits of integration, first draw a picture of what you're looking at. Your radius will go from 0 to z, your theta will go from 0 to 2pi, and your z will go from 0 to 1.