# Homework Help: Finding the area of a surface

1. Jun 5, 2010

1. The problem statement, all variables and given/known data
Find the area of the surface: The part of the plane 3x + 2y + z = 10 that lies in the first octant

2. Relevant equations
$$A=\iint_{D}|\vec{r}_{u}\times\vec{r}_{v}|dA$$

3. The attempt at a solution
I parametrized the plane:

$$\vec{r}\left(u,v\right)=<u,v,6-3u-2v>$$

and found the partial derivatives of u and v
$$\vec{r}_{u}=<1,0,-3>$$

$$\vec{r}_{v}=<0,1,-2>$$

then got this for their cross product:
$$\vec{r}_{u}\times\vec{r}_{v}=<3,2,1>$$

$$|\vec{r}_{u}\times\vec{r}_{v}|=\sqrt{14}$$

I think u goes from 0 to 2 because that's the highest and lowest values for u before the plane exits the first octant. This is also why I think the integral of v goes from 0 to 3. I obtain this answer:
$$\int_{0}^{2}\int_{0}^{3}\sqrt{14}dvdu=6\sqrt{14}$$

when the actual answer is half of that. What have I done wrong?

2. Jun 6, 2010

### tiny-tim

(I haven't ploughed through all your equations, but …)

shouldn't the u limits depend on v (or vice versa)?

3. Jun 6, 2010

### LCKurtz

And shouldn't that 6 be a 10?

4. Jun 6, 2010

Hi tiny-tim

I thought a surface in 3D was supposed to be parameterized with 2 variables u and v? (whereas a line integral in 2D was parameterized with just one variable t)

Sorry, I copied the question down wrong :yuck:

It should be 3x + 2y + z = 6. Don't know where I got that 10 from.

5. Jun 7, 2010

### LCKurtz

That's right. But your u and v are just new names for x and y. If you draw the first octant portion of the plane by locating its three intercepts, you will see the (x,y) domain is a triangle. Your limits on your integral describe a 2 by 3 rectangle.

6. Jun 7, 2010

Oh! I see. Since the domain is a triangle, its area is half of the rectangle I had integrated.

Thank you!

7. Jun 7, 2010

### tiny-tim

Hold it!

Yes, it is a triangle, and therefore half the area of the rectangle, but that's not what you're supposed to be learning from this question.

You need to understand how one of the limits of a double integral can depend on one of the variables, and you clearly don't understand that yet.

Go back to your professor (or your book) and find how to integrate over irregular areas.

8. Jun 7, 2010

I think I do understand

The domain is the triangle with vertices (2, 0), (0, 3), and (0, 0) in the xy plane; thus, I integrate like this

$$\intop_{0}^{2}\intop_{0}^{3-1.5x}\sqrt{14}dydx=\intop_{0}^{2}\sqrt{14}y|_{0}^{3-1.5x}dx=\sqrt{14}\intop_{0}^{2}3-1.5xdx=\sqrt{14}\left[3x-0.75x^{2}\right]_{0}^{2}$$

$$=\sqrt{14}\left(6-3\right)=3\sqrt{14}$$

and get half the answer I first got. It's just that seeing it as a triangle with half the area of the rectangle was a lazy way to understand what I did wrong

9. Jun 7, 2010

### LCKurtz

And hopefully you realize that had there been a variable density involved, taking half the answer over the rectangle would most likely have given the wrong answer. Dangerous shortcut.

10. Jun 7, 2010