Integrals with curl dot products

[clairez93]

Homework Statement

1. Evaluate \int_{S}\int curl F \cdot N dS where S is the closed surface of the solid bounded by the graphs of x = 4, z = 9 - y^2, and the coordinate planes.

F(x,y,z) = (4xy + z^2)i + (2x^2 + 6y)j + 2xzk

2. Use Stokes's Theorem to evaluate \int_{C}F\cdot T dS

F(x,y,z) = xyzi + yj +zk
S: 3x+4y+2z=12, first octant

Homework Equations

The Attempt at a Solution

1. For this one, I found the curl to be -6yi. However, I am at a loss as to how to get the N dS part without some sort of given equation for S? The book answer is 0.

2.
First I found the curl to be:
xyj - xzk

I then used a theorem in my book to find N ds:
3/2i + 2j + k

Then I took the dot product:
<0, xy, -xz> \cdot <\frac{3}{2}, 2, 1> = 2xy - xz

Integrating:
\int^{4}_{0}\int^{4-\frac{4y}{3}}_{0}(2xy-x(-\frac{3x}{2} - 2y + 6)*dx*dy
which comes out to 64/27.

The book answer is 0.

Any pointers as to what I'm doing wrong would be appreciated.

 

[gabbagabbahey]

1. For this one, I found the curl to be -6yi. However, I am at a loss as to how to get the N dS part without some sort of given equation for S? The book answer is 0.

You can either use Stokes' theorem for this, and the answer will be immediate; or you can find the equation of the surface that bounds the volume described in the question and integrate directly. (If you are going to use the second method, I recommend you break the surface into 4 separate surfaces to make it easier, and start by sketching the volume so you can see what I mean)

then used a theorem in my book to find N ds:
3/2i + 2j + k

Shouldn't \textbf{n}dS be a differential vector?