Homework SourseEvaluate the value enclosed by the paraboloid zx2y2 and the
Evaluate the value enclosed by the paraboloid z=(x^2)+(y^2) and the xy-plane over the cylinder 2x=(x^2)+(y^2)
Evaluate the value enclosed by the paraboloid z=(x^2)+(y^2) and the xy-plane over the cylinder 2x=(x^2)+(y^2)
Solution
We use spherical coordinates:
Z 2 0 Z 0 (x 2 + y 2 + z 2 )a 2 sin d d = Z 2 0 Z 0 (a 2 )a 2 sin d d = Z 2 0 a 4 cos 0 d = Z 2 0 2a 4 d = 4a4 . (b) Use symmetry and part (a) to easily find R R S y 2 dS. Solution: We discovered that 4a4 = R R S x 2+y 2+z 2 dS = R R S x 2 dS + R R S y 2 dS + R R S z 2 dS. Since the sphere is symmetric, all three integrals are the same. So 4a4 = 3 R R S y 2 dS, and R R S y 2 dS = 4 3 a4 .
