calculus III

[froznkiwi]

I have a few questions. I have an answer for the first one, but I don't know what to do with the other two. Can you verify my answer and help me with the others?

Thank you,
Jenny

∫ ∫ (1 + 2x) dA, where R is the region bounded by x = y² and x – y = 2
R


ANS : 0.6


Determine the volume of the part of the cylinder x² + y² - 2x = 0 that lies in the sphere x² + y² + z² = 4.


Find the volume of the ellipsoid
((x-2)²/4) + ((y+1)²/4) + ((z-4)²/16) = 1



Thanks again!

 

[pka]

You may want to rethink the first problem.

 

[soroban]

Hello, Jenny!

I think I've figured out the last one . . .

Find the volume of the ellipsoid
(x-2)²/4 + (y+1)²/4 + (z-4)²/16 = 1

The ellipsoid has its center at (2,-1,4).
It has an "x-radius" of 2, a "y-radius" of 2, and a "z-radius" of 4.

Since we are interested in its volume only, we can translate it to the Origin:
. . . x²/4 + y²/4 + z²/16 .= .1

Since the ellipsoid is symmetric to the three axes and the three coordinate planes,
. . we can find the volume in the first octant, and multiply by 8.
. . . . . . . . . . . . . . . . . ________
Solve for z: . z . = . 2√4 - x² - y²

At z = 0 (the xy-plane), we have the circle: . x² + y² = 4
. . . . . . . . . . . . ____
. . . or: . y .= .√4 - x²

. . . . . . . . . . . . . . . . . . . . . . . . . . .2 .√{4-x²}
I believe the volume is: . V . = . 8 ∫ . ∫ 2√{4 - x² - y²} dy dx
. . . . . . . . . . . . . . . . . . . . . . . . . . .0 . 0

Someone check my reasoning and my work . . . please!