Evaluate

where

is the region enclosed by the planes and and by

the cylinders and .

- March 4th 2011, 03:13 AMmaximus101double integration for a region W which is enclosed by planes and cylinders given
Evaluate

where

is the region enclosed by the planes and and by

the cylinders and . - March 4th 2011, 03:39 AMCaptainBlack
- March 4th 2011, 03:44 AMProve It
- March 5th 2011, 11:25 AMmaximus101
- March 5th 2011, 11:26 AMmaximus101
- March 5th 2011, 11:53 AMDSYEAYAssistance
Anyway, cylindrical coordinate is very much similar to polar coordinate except we have a z component thats all.

y=

x=

Rmb to include a conversion factor or (r) into the integrand. - March 5th 2011, 11:57 AMAllanCuz
Well, we can certainly switch to polar cordinates and find our bounds.

Polar cordinates are defined as,

Our integral is then,

The above was just to show you how everything comes together. To make it a little more cleaner.

Where A represents the area of our region bounded in the x-y domain. Certainly this happens between the 2 circles, that is

and

Thus,

Note that r represents the distance of the radial arm sweeping out from the origin. Think of it a circle with radius P which is centered at the origin. If we draw a line from the origin to the bounds of the circle, which would be P, we can obtain the area of the circle by rotating that radial arm, which we call R by 360 degrees. And that's what we're really going for, the area!

This topic actually requires a little more detail so I reccomend you consult your text about polar coordinates or look at the introduction to multiple integration stickied at the top of this forum!

But back to the question, we have defined our radial arm (R) but we need to sweep it across the domain to cover the entire area. In our case we have no restrictions so we are sweeping that arm from 0 to 360 degrees. We represent this by,

Combining this result with what we wrote above,

Of course we can re-arrange our integral,

You should be able to evaluate this now :) - March 6th 2011, 03:12 PMmaximus101
(Happy)got it, thank you very much