# Homework Help: Integrals with curl dot products

1. May 1, 2010

### clairez93

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. 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.

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

2. May 1, 2010

### gabbagabbahey

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)

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