- #1
LunaFly
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Homework Statement
Evaluate the surface integral:
∫∫s y dS
S is the part of the paraboloid y= x2 + z2 that lies inside the cylinder x2 + z2 =4.
Homework Equations
∫∫sf(x,y,z)dS = ∫∫Df(r(u,v))*|ru x rv|dA
The Attempt at a Solution
I've drawn the region D in the xz-plane as a circle with radius 2. My limits of integration are r between 0 and 2, and t between 0 and 2∏.
I've parametrized the surface as:
r(r,t)= <rcos(t), r2, rsin(t)>
I've gotten the vectors:
rr=<cos(t), 2r, sin(t)>
rt=<-rsin(t), 0, rcos(t)>
Trying to find dS, I did:
rr X rt = <2r2cos(t), -r, 2r2sin(t)>,
and the magnitude of that is r√(4r2 +1).
Overall, I end up with dS = r2*√(4r2+1)drdt.
The integral becomes:
∫∫r4*√(4r2+1)drdt, with limits as mentioned above.
However this is not the correct answer. I am having a difficult time with these surface integrals and I am not really sure why. I keep getting confused about whether or not I am parametrizing
the surfaces correctly, so that may be where my error is. Any help would be much appreciated! Thanks
-Luna
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