Using double integration to find area of a cylinder

[liishii]

Homework Statement

Find the area of that portion of a cylinder x^2+y^2=1 lying above the xy-plane and below the plane z=3y.

Homework Equations

dr=sqrt(1+f_x²+f_y2[/SUB])dxdy

or

dr=|r_u x r_v| dudv

The Attempt at a Solution

I attempted to parametrize the function in terms of theta (t) and z. I got r(t,z)=icost+jsint+kz. Then, I found the partials with respect to t and z and got: r_t=-sinti+costj and r_z=z, where t=0 to 2pi and z=0 to 3y=3sint.
I took the cross product of the two partials to get icost+jsint+0k. I used the cross product to find |r_t x r_z|=1. Then, plugging this into the second equation above, I got the answer to be 0, which is incorrect. What am I doing wrong?? Thanks for your help!

 

[vela]

Your limits for t are wrong. Remember, you only want the part above the xy-plane.

 

[liishii]

oh that makes sense! so the t limits had to be from 0 to pi, not 2pi.
I got the right answer! Thanks so much vela!