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Triple Integrals with Cylindrical Coordinates

  • #1

Homework Statement


Evaluate the integral, where E is the solid in the first octant that lies beneath the paraboloid z = 9 - x2 - y2.

∫∫∫(2(x^3+xy^2))dV

Homework Equations



x=rcosθ
y=rsinθ
x^2+y^2=r^2

The Attempt at a Solution



θ=0 to 2π, r=0 to 3, z=0 to (9-r^2)

2(x^3+xy^2)=2x(x^2+y^2)=2rcos(θ)(r^2)

∫0 to 2π ∫0 to 3 ∫0 to (9-r^2) (2rcos(θ)r^2)rdzdrdθ

I was wondering if my bounds were correct. And when I solved the integral I keep getting an answer of 0, which is incorrect. Can someone please help me with this problem?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
1,796
53
If you're integrating over the first octant, then theta doesn't go from 0 to 2pi. Other than that, the rest is ok.
 
  • #3
Thank you so much! That was the problem. I missed the part where it said the first octant.
 

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