Question RE: stoke's theorem. difficulty finding limits of integration

In summary, the surface integral for the vector field <2z, x, 1> over the surface S, which is a section of a plane above a region in the xy plane, can be evaluated using Stoke's theorem. The curl of the vector field is <0,2,1> and the normal to the surface is <-4,8,1>. The integrand in the area formula is sqrt(1+4^2+8^2), a constant. Multiplying this by the constant obtained from Stoke's theorem and the area of the surface, which is a section of a circle, we get the final answer for the surface integral.
  • #1
wown
22
0

Homework Statement


Let S be the part of the plane z=f(x,y)=4x - 8y +5 above the region (x-1)^2 + (y-3)^2 <= 9 oriented with an upward pointing normal. Use Stoke's theorem to evaluate the surface integral for the vector field <2z, x, 1>.



Homework Equations


Stoke's Theorem is surface integral of Fdot product ds = surface integral curl(F) dot product dS


The Attempt at a Solution


I figured out the curl: <0,2,1>. I am having trouble with finding the limits of integration for the normal to the surface. I see that they are asking for the surface that lies on the plane z=4x-8y+5 above the xy plane. Great.

the normal to the surface is easy as well : <-4, 8, 1>. here is where i hit a stop.
reason 1: there are no variables in either of the two vectors and this flags an alarm because I feel I did something wrong.
reason 2: I cannot figure out the limits of integration to the surface. I considered polar/spherical coordinates but since the boundary on the plane is an ellipse, these do not work (unless there is a way to make them work, in which case I do not know what that is).

One thing i considered doing was reasoning that since stoke's theorem depends solely on the boundary, I said "well we basically have a boundary of half an ellipse in the xy plane". So I solve 4x-8y+5=0 for x or y, plug that into the equation for the original circle with radius 3, and solve for the extreme values of x and y. This way I can come up with the equation for an ellipse and set one of the limits equal to the extreme values and the other limit to the upper half of the ellipse... Except that sounds way too complicated for the average problem in my class and I feel that I must have missed something.

Any guidance would be much appreciated. Thanks!
 
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  • #2
The dot of the curl and the normal is a constant. That's a good thing. So to integrate over the surface you just need to multiply it by the area of the surface. The surface is a section of a plane over a circle of radius 3 in the x-y plane. Can you think of a way to do that?
 
  • #3
right. but figuring out the area of the surface is where i am having most difficulty :(

the plane cuts through the circle when 0 = 4x - 8y +5. So I can solve for y and draw the line where the circle is cut. I also know that z= 9 when x = 1 and y = 0 and 9 is the highest value of z.

where do i go from there? How can i calculate the area of the part of the ellipse on the plane?
 
  • #4
f(x,y)=4x - 8y +5. What's the expression in the area formula you need to integrate in the x-y plane to get the area of that surface?
 
  • #5
I really have no clue :(
have you drawn the surface? it is a very weird looking thing. the projection of the surface on the xy plane is of course the part of the circle with radius 3 and center (1,23) that is cut by the line 4x-8y+5 and lies to the left of that line. The surface itself has one edge that has the line 4x-8y+5 and has max height of z=9 when x=1, y=0. so it is an ellipse with a ray that doesn't run through the middle and is slanted.
 
  • #6
I haven't really 'drawn' it. As I'm visualizing it, it's just a slanted ellipse. Could you answer my question about what function you should integrate over the circle in the x-y plane to get the area of the ellipse? The details of the ellipse really aren't that important.
 
  • #7
hm pi*a*b? or in this case 1/2*pi*(x or y)*z(which is 4x-8y+5)?
 
  • #8
wown said:
hm pi*a*b? or in this case 1/2*pi*(x or y)*z(which is 4x-8y+5)?

Look up a formula for the area of a surface z=f(x,y). If you haven't done that yet, you may be worrying about details of the surface that you don't have to.
 
  • #9
OOOOOOOOOOOOOOOO lol it just clicked... i really should not be doing these problems after working 10 hour days :)

but to make sure then, i would have to set 4x-8y+5 equal to the equation of the circle to figure out the limits on x and y, right?
 
  • #10
wown said:
OOOOOOOOOOOOOOOO lol it just clicked... i really should not be doing these problems after working 10 hour days :)

but to make sure then, i would have to set 4x-8y+5 equal to the equation of the circle to figure out the limits on x and y, right?

You haven't answered my question yet. You don't NEED any explicit limits. What's the integrand in the area formula??
 
  • #11
it would be the integral of f(x,y,g(x,y))*sqrt(1+4^2 (this the derivative of g in terms of x) + 8^2))dxdy
 
  • #12
wown said:
it would be the integral of f(x,y,g(x,y))*sqrt(1+4^2 (this the derivative of g in terms of x) + 8^2))dxdy

I think it's just sqrt(1+4^2+8^2). It's a constant. What's the f(x,y,g(x,y)) thing doing in there? http://mathworld.wolfram.com/SurfaceArea.html formula 3.
 
  • #13
first, thanks so much for bearing with me and being so patient :)

so, basically i have curl*normal = <0,2,1>dot prod<-4,8,1> = 17 = f(x,y,g(x,y)) which is a constant. multiply this by the sqrt of (1+16+64) = 9. So the final is a constant = 153 which i multiply times the area of the circle = pi*9 and that is my final answer?
 
  • #14
Dick said:
I think it's just sqrt(1+4^2+8^2). It's a constant. What's the f(x,y,g(x,y)) thing doing in there? http://mathworld.wolfram.com/SurfaceArea.html formula 3.

hm interesting. my calc book includes the f(x,y,g(x,y))... which in this case would make sense, no? because ultimatwly i need to multiply by the constant i got from stoke's formula
 
  • #15
wown said:
it would be the integral of f(x,y,g(x,y))*sqrt(1+4^2 (this the derivative of g in terms of x) + 8^2))dxdy

wown said:
first, thanks so much for bearing with me and being so patient :)

so, basically i have curl*normal = <0,2,1>dot prod<-4,8,1> = 17 = f(x,y,g(x,y)) which is a constant. multiply this by the sqrt of (1+16+64) = 9. So the final is a constant = 153 which i multiply times the area of the circle = pi*9 and that is my final answer?

Well, I'm with you as far as you multipy sqrt(1+16+64) times the area of the circle to find the area of the slanted ellipse. And sure, then you multiply by curl.normal to get the final answer. I'll let you actually check the number. But the point is so many things are constant you don't need to worry about messy details like limits.
 
  • #16
wown said:
hm interesting. my calc book includes the f(x,y,g(x,y))... which in this case would make sense, no? because ultimatwly i need to multiply by the constant i got from stoke's formula

Mmm. I really don't know what the book is talking about. What's g(x,y) supposed to be?
 

1. What is Stokes' Theorem and why is it important?

Stokes' Theorem is a mathematical theorem used in vector calculus that relates the surface integral of a vector field to the line integral of its curl. It is important because it provides a way to calculate the circulation of a vector field over a closed surface without having to directly integrate the field over the surface.

2. How do you apply Stokes' Theorem to find the limits of integration?

To apply Stokes' Theorem and find the limits of integration, you first need to determine the orientation of the surface and the direction of the curve. Then, you can use the parametric equations of the curve to define the limits of integration for the line integral. The surface integral will have limits that correspond to the projection of the curve onto the surface.

3. What are some common difficulties when using Stokes' Theorem to find limits of integration?

One common difficulty is determining the correct orientation of the surface and direction of the curve. Another difficulty can arise when the surface is not explicitly given and needs to be determined from a given boundary curve. Additionally, finding the parametric equations for the curve and the projection onto the surface can also be challenging.

4. How do you know if you have the correct limits of integration when using Stokes' Theorem?

You can check if you have the correct limits of integration by making sure that they correspond to the boundaries of the surface and the projection of the curve onto the surface. Additionally, you can also verify that the orientation of the surface and direction of the curve are consistent with the given vector field.

5. Are there any tips for finding the limits of integration when using Stokes' Theorem?

One tip is to draw a diagram of the surface, curve, and projection to help visualize the problem. Another tip is to carefully consider the orientation and direction, making sure they are consistent with the given vector field. It can also be helpful to break the surface and curve into smaller, simpler pieces to make the calculation more manageable.

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