# Thread: Need Help with vector calculus problems!

1. ## Need Help with vector calculus problems!

1) Use the triple integral to find the volume of the solid bounded above and below by the sphere x^2 + y^2 + z^2 = 9 and cylinder x^2 + y^2 = 4

2) Find the work done by a force F(x,y,z) = (x+y)i + (xy)j - (z^2)k acting on a particle that moves along the line segment from (0,0,0) to (1,3,1) and then along the line segment from (1,3,1) to (2,-1,4)

3) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off the by the plane z=4. If F(x,y,z) = (x^2 + z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S.

If you could help me with any of these problems much would be appreciated. If you could show your work then that would be great too! Thanks!

2. Originally Posted by davidson89
1) Use the triple integral to find the volume of the solid bounded above and below by the sphere x^2 + y^2 + z^2 = 9 and cylinder x^2 + y^2 = 4

[snip]
1) Integrate in the order dz dx dy.

z-integral terminals: Lower terminal is z = - sqrt{9 - x^2 - y^2} (lower surface). Upper terminal is z = sqrt{9 - x^2 - y^2} (upper surface).

The double integral dx dy is over the region of the xy-plane defined by x^2 + y^2 = 4. I'm sure you can set up the integral terminals. Personally, I'd switch to polar coordinates once the z-integration has been done .....

Note: You can use the symmetry of the problem .... Two times the triple integral where the lower terminal is z = 0 (lower surface) and the upper terminal is z = sqrt{9 - x^2 - y^2} (upper surface) ......

3. Originally Posted by davidson89
[snip]

2) Find the work done by a force F(x,y,z) = (x+y)i + (xy)j - (z^2)k acting on a particle that moves along the line segment from (0,0,0) to (1,3,1) and then along the line segment from (1,3,1) to (2,-1,4)
[snip]
2) You need to evaluate the line integral int F.dr over the two line segments.

Line segment from (0, 0, 0) to (1, 3, 1):

The parametric equation of the line segment is x = t, y = 3t, z = t where t = 0 to t = 1.

So the line integral becomes

int[t = 0, t = 1] [(t + 3t)i + (3t^2)j - (t^2)k].[1i + 3j + 1k] dt

(you know where this expression has come from, right?)

Line segment from (1,3,1) to (2,-1,4):

Get the parametric equation and proceed in a similar way to the above.

4. Originally Posted by davidson89
[snip]

3) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off the by the plane z=4. If F(x,y,z) = (x^2 + z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S.

If you could help me with any of these problems much would be appreciated. If you could show your work then that would be great too! Thanks!
More definitions. Where are you stuck here? Have you drawn a picture? Can you set up the flux integral? Can you do the integrations?

5. Hey, thanks for your help. Yes i am having trouble setting up the integrals for 1 and 3. I cant seem to figure out the conditions for the integrals. Also, i really do not have an idea to the flux problem, your clarification would be much appreciated on how to do them

6. Originally Posted by davidson89
Hey, thanks for your help. Yes i am having trouble setting up the integrals for 1 and 3. I cant seem to figure out the conditions for the integrals. Also, i really do not have an idea to the flux problem, your clarification would be much appreciated on how to do them
1) V = int int int dx dy dz and I've already given the integral terminals for this triple integral as well as advice on how to approach the integrations.

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For 3), you must at least have the definition Flux = int int F . dS in your notes or textbook .....

The surface is the paraboloid z= x^2 + y^2. Use this equation to get an expression for vector dS in terms of dx dx (your notes and/or textbook should have the necessary formula for doing this):

vector dS = (.........) dx dy

where you get to fill in the dotted line.

Now do the dot product F.(.......). This converts the flux integral into a routine double integral in the xy-plane.

The region of integration in the xy-plane is the circle x^2 + y^2 = 4 (substitute z = 4 into z = x^2 + y^2). This suggests converting to polar coordinates (for an easier calculation) to evaluate the double integral.

7. hey so for number 2 would the integration be:

integral from 0 to 1 [ (1+t+3-4t) + (1+t)(3-4t)(-4) - (1+3t)^2(3)]

i get the answer of -139/6

If i add this with the result of the first integration, i get -37/2 is this correct

8. Originally Posted by davidson89
hey so for number 2 would the integration be:

integral from 0 to 1 [ (1+t+3-4t) + (1+t)(3-4t)(-4) - (1+3t)^2(3)]

i get the answer of -139/6

If i add this with the result of the first integration, i get -37/2 is this correct
The set up of the line integral along the second line segment looks fine. I haven't bothered to check the value.

9. So i dont get for number 3 how to set up the integral.... if it is in the order dx dy dz, then the integral terminals are still
z = 0 to z = (4 - x)/2
x = 0 to x = 4
y = 0 to y = 2. ?

So it would be

int(0 to 2) int(0 to 4) int(0 to 4-x/2) [ dx dy dz ]

or should it be

int(0 to 4-x/2) int (0 to 2) int (0 to 4) [ dx dy dz]

could it be done like this?

int (0 to 4) int (0 to 2) int(0 to (4-x)/2) [ dz dy dx]

10. Originally Posted by davidson89
[snip]
3) Let S be the first-octant portion of the paraboloid z= x^2 + y^2 that is cut off the by the plane z=4. If F(x,y,z) = (x^2 + z)i + (zy^2)j + (x^2 + y^2 + z)k, find the flux of F through S.

If you could help me with any of these problems much would be appreciated. If you could show your work then that would be great too! Thanks!
Originally Posted by davidson89
So i dont get for number 3 how to set up the integral.... if it is in the order dx dy dz, then the integral terminals are still
z = 0 to z = (4 - x)/2
x = 0 to x = 4
y = 0 to y = 2. ?

So it would be

int(0 to 2) int(0 to 4) int(0 to 4-x/2) [ dx dy dz ]

or should it be

int(0 to 4-x/2) int (0 to 2) int (0 to 4) [ dx dy dz]

could it be done like this?

int (0 to 4) int (0 to 2) int(0 to (4-x)/2) [ dz dy dx]

I don't see how this relates to Q3 as posted. I have discussed Q3 in post #6. What part of that discussion don't you understand.

11. I am sorry i meant to post this reply in the other post. Haha My question was how to set up the integral for this problem

Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S

12. Originally Posted by davidson89
I am sorry i meant to post this reply in the other post. Haha My question was how to set up the integral for this problem

Let S be the surface of the region bounded by the coordinate planes and the planes x + 2z = 4 and y = 2. Use the Divergence Theorem to the flux of F(x,y,z) = (2xz)i + (xyz)j + (yz)k through S

do you not understand regarding this question.

Seriously, I'm struggling to keep a track of all your questions. It's just getting too confusing for me. In the future:

1. Present one question per thread. Not several.

2. When a reply is given, ask follow-up questions at that thread if you still don't understand. If you post the same question again in other places, people just end up saying the same things several times which wastes their time.

Go back. Read the replies I have given to each of your questions. Then attempt to understand how to do the question. If you're still stuck after doing these things, state clearly where you're stuck.

Many of your problesm look like they could be resolved if you went back and revised the relevant mathematics in your textbook and/or class notes.