# Surface integral from vector calculus

## Homework Statement

F = (x^2) i + (y^2) j + (z) k, S is the cone z = (x^2+y^2) ^ (1/2), with x^2 +y^2 <= 1, x >= 0, y >= 0, oriented upward.

All of the above

## The Attempt at a Solution

My attempted solution is 0. But other students claim that the answers is (2/3)pi.

My work is this:
double integral ( (x^2) i + (y^2) j + (z) k ) . ( -(x / (x^2+y^2)^ (1/2) i -(y / (x^2+y^2)^ (1/2) j + (x^2 + y^2) ^ (1/2) k ) dx dy

Then finding common denominator and simplifying:

double integral ( (- x - y ) / (x^2+y^2)^(1/2) ) dx dy

Changing to polar coordinates:

(first integral 0 to 2pi)(second integral 0 to 1) (-cos(theta) - sin(theta) / (r^2) ) * rdrd(theta)

Finally, doing those integrals I have the result of 0.

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HallsofIvy
Homework Helper
I do not get 0 as the integral.

tiny-tim
Homework Helper
Welcome to PF!

Hi m453438! Welcome to PF! (have a theta: θ and a pi: π and a √ and a ∫ and a ≤ and ≥, and try using the X2 tag just above the Reply box )

Useful tip: when you do it again, try putting r = √(x2 + y2) at the start

it makes it a lot easier to write, which means you're less likely to make a mistake. 