- #1

- 100

- 0

## Homework Statement

Compute ∫∫

_{S}

**F**.d

**S**

**F**(x,y,z)=<y,x

^{2}y,e

^{xz}> over x

^{2}+y

^{2}=9, -3<=z<=3, outward pointing normal.

## The Attempt at a Solution

I parameterized the surface in cylindrical coordinates:

**Φ**(z,θ)=<3cosθ,3sinθ,z>.

The normal vector of this surface is

**n**(z,θ)=<0,0,1>x<-3sinθ,3cosθ,0>=

<-3cosθ,-3sinθ,0>.

∫∫

_{S}

**F**.d

**S**

=∫∫

**F**(

**Φ**(z,θ)).

**n**(z,θ)dzdθ

**F**(

**Φ**(z,θ))=<3sinθ,9cos

^{2}θ,e

^{3zcosθ}>

I took the dot product of that and the normal, then evaluated the integral with z from -3 to 3 and θ from 0 to 2

**π**and got an answer of 0. I know the answer should be (243

**π**)/2

Some help in finding my mistake would be much appreciated. And I know it wasn't from evaluating the integral, as I used my calculator(we are allowed to).