# Surface integral questions

#### godiva

I need to find the surface integrals for these various questions. Some I don't know where to begin, others I don't know how to continue. Help would be appreciated  I get: $$\displaystyle \int \int F \cdot n dS$$
$$\displaystyle F = (x,y,2z)$$
$$\displaystyle n = \frac {2x}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {2y}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {1}{\sqrt {4x^2 + 4y^2 + 1}}$$
$$\displaystyle \int \int F \cdot n dS = \int \int 2 dA$$

But not sure what the limits are to proceed I'm guessing similar manner to question above, but do I get z on its own i.e. $$\displaystyle z = 3-3x-\frac {3}{2}y$$ then proceed. But once again, not sure what the limits would be Using divF = 3 then $$\displaystyle \int^1_0 \int^1_0 \int^1_0 3 dzdxdy = 3$$ Is that right? Need to use divergence again, but not sure how to proceed. I could do the same as below, but not sure how to proceed I got the curl to be i + j + k and $$\displaystyle \sqrt { ( \frac {\partial f}{\partial x})^2 + ( \frac {\partial f}{\partial y})^2 + 1 } = \sqrt {3}$$
Not sure how to proceed

#### Bruno J.

MHF Hall of Honor
I didn't read all your problems - way too many in one post! But I can tell you the first one is easily solved by the divergence theorem. Find $$\displaystyle \mbox{div } F$$ and integrate $$\displaystyle \mbox{div }F\ dV$$ inside the cube.

#### 11rdc11

I need to find the surface integrals for these various questions. Some I don't know where to begin, others I don't know how to continue. Help would be appreciated  I get: $$\displaystyle \int \int F \cdot n dS$$
$$\displaystyle F = (x,y,2z)$$
$$\displaystyle n = \frac {2x}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {2y}{\sqrt {4x^2 + 4y^2 + 1}} + \frac {1}{\sqrt {4x^2 + 4y^2 + 1}}$$
$$\displaystyle \int \int F \cdot n dS = \int \int 2 dA$$

But not sure what the limits are to proceed I'm guessing similar manner to question above, but do I get z on its own i.e. $$\displaystyle z = 3-3x-\frac {3}{2}y$$ then proceed. But once again, not sure what the limits would be Using divF = 3 then $$\displaystyle \int^1_0 \int^1_0 \int^1_0 3 dzdxdy = 3$$ Is that right? Need to use divergence again, but not sure how to proceed. I could do the same as below, but not sure how to proceed I got the curl to be i + j + k and $$\displaystyle \sqrt { ( \frac {\partial f}{\partial x})^2 + ( \frac {\partial f}{\partial y})^2 + 1 } = \sqrt {3}$$
Not sure how to proceed

for the second problem change it into polar and use the limits

$$\displaystyle \int^{1}_{0}\int^{2\pi}_{0}d\theta dr$$

for the 3rd one the limits are

$$\displaystyle \int^{\frac{3}{2}}_{0} \int^{-2x+3}_{0} dydx$$

Correct on 4

For question 5 find the divergence and then use

$$\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} drd\theta dz$$

Last edited:

#### godiva

For question 5 find the divergence and then use

$$\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} drd\theta dz$$
Thanks. For the above would an r need to be added? I remember something like that when i did coordinates, i.e. should it be $$\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} r drd\theta dz$$

Also, any ideas for the final 2? I've gone through my book twice and it doesn't have a proper example

#### 11rdc11

Thanks. For the above would an r need to be added? I remember something like that when i did coordinates, i.e. should it be $$\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} r drd\theta dz$$

Also, any ideas for the final 2? I've gone through my book twice and it doesn't have a proper example
Yes for the 6th one remember what Stokes theorem says. It does not matter what surface you use as long as C is the same. In this case it be alot easier to use the surface of z=1 with the domain being projected to the xy plane and you get the integral

$$\displaystyle \int^{1}_{0} \int^{2\pi}_{0} d\theta dr$$

Another way to do 6 is let

$$\displaystyle r(t)= \cos{t}i +\sin{t}j + k$$

$$\displaystyle f(r(t)) = \cos^2{t}i + \sin^2{t}j +k$$

$$\displaystyle r'(t) = -\sin{t}i +\cos{t}j + 0k$$

Now just evaluate the line integral

#### 11rdc11

Thanks. For the above would an r need to be added? I remember something like that when i did coordinates, i.e. should it be $$\displaystyle \int^{1}_{0} \int^{2\pi}_{0} \int^{a}_{0} r drd\theta dz$$

Also, any ideas for the final 2? I've gone through my book twice and it doesn't have a proper example

And for the last one I got a different curl than you did.

Mine was

$$\displaystyle -i +j -k$$

so

$$\displaystyle \int \int [-Mf_x - Nf_y +P]dxdy$$

f_x = -1

f_y = -1

$$\displaystyle \int^{1}_{0} \int^{1-x}_{0} dydx$$