Stokes Theorem paraboloid intersecting with cylinder

[gtfitzpatrick]

Homework Statement

Use stokes theorem to elaluate to integral \int\int_{s} curlF.dS where F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k and s is the part of the paraboliod z=x^2+ y^2 that lies inside the cylinder x^2 +y^2 =4 and is orientated upwards

Homework Equations

The Attempt at a Solution

so i use Stokes theorem \int\int_{s} curlF.dS = \oint_{c} F.dv

so i want to get parametric equations for C and so i get x=2cost y=2sint and z=4 i came up with these as the boundary curve c is a circle of raadius 2 and my z value of 4 because that is where the parabaloid and cylinder intersect...am i right in my thinking here?

so then dx= -2sintdt ; dy=2costdt and dz=0

so then i get \oint_{c} F.dv = \oint_{c} x^2 z^2 dx - y^2z^2 dy + xyzdz

which becomes \int^{2\pi}_{0} ((4cos^2 t )(16)(-2sint) - (4sin^2t)(16)(ccost)) dt

\int^{2\pi}_{0} ((-128cos^2 t )(sint) - (128sin^2 t)(cost)) dt

am i working on the right lines here?

 

[gtfitzpatrick]

whoops hold up a sec...not finished putting up my attempt, I am editing the post!

 

[gtfitzpatrick]

the original post is complete now. anyone got any ideas?

 

[gtfitzpatrick]

anyone?

 

[DryRun]

I think the last part is wrong and it should be:
$\int^{2\pi}_{0} (-128cos^2 t sint + 128sin^2 tcost) .dt$

 

[LCKurtz]

Homework Statement

Use stokes theorem to elaluate to integral \int\int_{s} curlF.dS where F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k and s is the part of the paraboliod z=x^2+ y^2 that lies inside the cylinder x^2 +y^2 =4 and is orientated upwards

Homework Equations

The Attempt at a Solution

so i use Stokes theorem \int\int_{s} curlF.dS = \oint_{c} F.dv

so i want to get parametric equations for C and so i get x=2cost y=2sint and z=4 i came up with these as the boundary curve c is a circle of raadius 2 and my z value of 4 because that is where the parabaloid and cylinder intersect...am i right in my thinking here?

so then dx= -2sintdt ; dy=2costdt and dz=0

so then i get \oint_{c} F.dv = \oint_{c} x^2 z^2 dx - y^2z^2 dy + xyzdz

Check that - sign.

which becomes \int^{2\pi}_{0} ((4cos^2 t )(16)(-2sint) - (4sin^2t)(16)(ccost)) dt

\int^{2\pi}_{0} ((-128cos^2 t )(sint) - (128sin^2 t)(cost)) dt

am i working on the right lines here?

Looks good except for that minus sign.

 

[DryRun]

The integration can be tricky but there is a simple trick to it. Let us know how it goes.

However, i have a question myself, since I'm still learning. I suppose it is OK for me to ask here? What if the orientation of the surface S was downwards? Then, how would the calculations from post #1 change?

 

[gtfitzpatrick]

Hi LCkurtz and Sharks
Thanks a million for comments.

To integrate i used substitution u=cost du=-sintdt and similar for sin.

as in \int cos^2 t sint =\frac{1}{3} cos^3 t i think this works

so i get \frac{128}{3} (cos^3 t + sin^3 t)^{2\pi}_{0}

=-\frac{128}{3} i hope :)

 

[gtfitzpatrick]

However, i have a question myself, since I'm still learning. I suppose it is OK for me to ask here? What if the orientation of the surface S was downwards? Then, how would the calculations from post #1 change?

i'm afraid i haven't a clue, sorry!

 

[DryRun]

But isn't the whole integral this:
$\int^{2\pi}_{0} (-128cos^2 t sint + 128sin^2 tcost) .dt$
Then, it would become:
$\int^{2\pi}_{0} (-128cos^2 t sint).dt + \int^{2\pi}_{0} (128sin^2 tcost).dt$

 

[LCKurtz]

The integration can be tricky but there is a simple trick to it. Let us know how it goes.

However, i have a question myself, since I'm still learning. I suppose it is OK for me to ask here? What if the orientation of the surface S was downwards? Then, how would the calculations from post #1 change?

I was tempted in my original post to ask the OP if he had thought about the orientation or whether he just took a guess. You use the right hand rule. In this case the normal was pointed upwards, so the right hand rule would give you counterclockwise in the xy plane as viewed from above which gives $t:\, 0\rightarrow 2\pi$. For downward orientation you go the other way around the circle which could be $t:\, 2\pi\rightarrow 0$. The effect is to change the sign of the answer.

 

[DryRun]

I was tempted in my original post to ask the OP if he had thought about the orientation or whether he just took a guess. You use the right hand rule. In this case the normal was pointed upwards, so the right hand rule would give you counterclockwise in the xy plane as viewed from above which gives $t:\, 0\rightarrow 2\pi$. For downward orientation you go the other way around the circle which could be $t:\, 2\pi\rightarrow 0$. The effect is to change the sign of the answer.

Thank you for this quick and accurate answer, LCKurtz.

 

[gtfitzpatrick]

Then, it would become:
$\int^{2\pi}_{0} (-128cos^2 t sint).dt + \int^{2\pi}_{0} (128sin^2 tcost).dt$

\int^{2\pi}_{0} (-128cos^2 t sint)dt + \int^{2\pi}_{0} (128sin^2 t cost)dt

=128[\frac{1}{3} cos^3 t ]^{2\pi}_{0} + 128[\frac{1}{3} sin^3 t ]^{2\pi}_{0}

=\frac{128}{3}[-1] = \frac{-128}{3}

 

[gtfitzpatrick]

I was tempted in my original post to ask the OP if he had thought about the orientation or whether he just took a guess. You use the right hand rule. In this case the normal was pointed upwards, so the right hand rule would give you counterclockwise in the xy plane as viewed from above which gives $t:\, 0\rightarrow 2\pi$. For downward orientation you go the other way around the circle which could be $t:\, 2\pi\rightarrow 0$. The effect is to change the sign of the answer.

ahhh thanks a million i had wondered about that alright. that cleared it up. thanks a million.

 

[DryRun]

\int^{2\pi}_{0} (-128cos^2 t sint)dt + \int^{2\pi}_{0} (128sin^2 t cost)dt

=128[\frac{1}{3} cos^3 t ]^{2\pi}_{0} + 128[\frac{1}{3} sin^3 t ]^{2\pi}_{0}

=\frac{128}{3}[-1] = \frac{-128}{3}

I don't think the evaluation part is correct.

 

[gtfitzpatrick]

=\frac{128}{3}[-1] = \frac{-128}{3}

whoops should be \frac{128}{3}[1-1] = 0

 

[gtfitzpatrick]

Thanks a million sharks! your brill!