Question

Using a polar double integral, find the volume in the first octant below the surface z=xy, inside cylinder r=4cos(theta), and between the cylinders x^2 + y^2=4 and x^2 + y^2=8.

Answer 1

in polar the three graphs are\[r=2\]\[r=2\sqrt2\]\[r=4\cos\theta\]can you find the intersection points in terms of theta?

Answer 2

1st octant...

Answer 3

Make the equations equal to one another...

Answer 4

yes, and what do you get for theta?

Answer 5

arccos 2^(1/2)/2

Answer 6

well that's one, what angle does that correspond to?
and what is the other intersection?

Answer 7

45 degrees

Answer 8

and 60 degrees

Answer 9

yes, 45 and 60, but you better use radians in calc

Answer 10

so which fuunction is the outer radius and which one the inner radius for\[0\le\theta\le\frac\pi4\]?

Answer 11

r=2 is inner and r=2sqrt2 is outer...

Answer 12

yes, good :)
and for the interval\[\frac\pi4\le\theta\le\frac\pi3\]what is it?

Answer 13

r=4cos (theta) is outer and r=2sqrt2 is inner...

Answer 14

I think you have the inner wrong

Answer 15

oh, inner is r=2

Answer 16

right
and what are your integrands?

Answer 17

I'm not too sure about that

Answer 18

you are integrating xy
sub in x=rcost, y=rsint

Answer 19

above t=theta in case you didn't guess
also you need to use dA for polar coordinate

Answer 20

so it'll be the double integral of r^2 cost sint...

Answer 21

that is the xy part, and what is dA in polar?