How Do I Correctly Calculate Volume for a UVZ Coordinate System?

[sedaw]

attached herewith : the problem with my attempt to solve it .

 

[HallsofIvy]

1. What is "an" in "cot ant"?

2. How did you get the limits of integration? Specifically, what are the limits of integration on u and v?

 

[sedaw]

1. What is "an" in "cot ant"?

2. How did you get the limits of integration? Specifically, what are the limits of integration on u and v?

cotan(t) = cos(t)/sit(t)

the limits of integration on u & v simply received from the projection of the region bounded by V on the UV plane -> you can see it circle with radius R=1 .

the limits of integration on z:

you can see in sketch : 0<=z<=U+V (of course just for u & v that in region of the projection )
TNX ...

 

[jackmell]

Yeah, the ant thing . . . not a good way to start sedaw. Did you just enter "cotant" in latex and it partitioned it that way? Just use $\cot$. But even worst, that picture looks nothing like what I think your V is. Guess the J is Jacobian. But I wouldn't try to change variables until I first tried to solve it the old-fashioned way: just use x, y, and z. Oh, but I would also try to plot a real-looking picture first. Is the volume that part under the paraboloid [itex]z=x^2+y^2[/tex] contained by the transparent "square-tube" of $x^4+y^4=1$ under the paraboloid in the plot below?<br /> <br /> If so, can you see how to arrive at:<br /> <br /> $$V=8\int_0^{1/2^{1/4}}\int_x^{\sqrt[4]{1-x^4}}\left(x^2+y^2\right) dydx$$<br /> <br /> That comes out to about 2.22. Would be nice if you already know the answer to see at least if this is on the right track and I realize that's a tough integral to solve symbolically, but it get's you grounded at least with something concrete to work with and then perhaps you can use that integral to then change variables. Maybe not though. It's just a suggestion.[/itex]

 

[sedaw]

ok.. thank you very much i am appreciate your help.
by the way my picture is V for the uvz axis and not xyz .