Hi, I need urgent help with these 3 integrals problems ... been stuck on the questions and the deadline is Friday. Thanks a lot !

 

 

1) For the green's theorem,

 

I got the answer : 27.552. Not sure whether it is correct. Please kindly explain in steps so I know where I went wrong.

 

 

2) I'm totally unsure about finding the surface integral. How do I know the shape of the surface ?

 

 

 

3) and finding the curl function f

 

All I know is, it has something to do with curl. Something like F = grad f. How do I find the function ?

 

 

Thanks again !

Homework problem?

 

1.

 

Apply green's theorem:

http://mathworld.wolfram.com/GreensTheorem.html

 

 

2.

the G = bit tells you the shape of the surface, what is dA in terms of x,y,z?

 

3.

Is there any more information given?

 

Take this info with a bit of sceptisism I need more sleep.

Hi, I need urgent help with these 3 integrals problems ... been stuck on the questions and the deadline is Friday. Thanks a lot !

 

I won't provide full solutions, but I will give you hints.

 

1) For the green's theorem,

 

I got the answer : 27.552. Not sure whether it is correct. Please kindly explain in steps so I know where I went wrong.

 

The hardest step here is a single application of Green's theorem. You should get that:

 

[math]\int_{C} F® \, dr = \int_0^3 \int_1^2\left( \frac{\partial}{\partial x} (ye^x) - \frac{\partial}{\partial y}(x \log y) \right) \, dy \, dx [/math]

 

(I would appreciate someone double checking this because it's been a long time since I've used Green's theorem). The double integral on the right hand side is easy to evaluate.

 

2) I'm totally unsure about finding the surface integral. How do I know the shape of the surface ?

 

Try to look at what you've got there. You should know that the equation [math]x^2+y^2 = a^2[/math] gives you a circle of radius a. [math]y \geq 0[/math] means that we're going to be reduced to half a circle. And [math]0 \leq z \leq h[/math] simply means that we take this half-circle and extend it upwards by length h. So, S is basically just a cylinder of height h which has been cut in half.

 

Hopefully this well help you visualise what's going on.

 

All I know is, it has something to do with curl. Something like F = grad f. How do I find the function ?

 

Well, this is a bit tricky but simple once you've worked out what to do. If we have a function [math]f(x,y)[/math], then

 

[math]\nabla f(x,y) = \left( \frac{\partial f}{\partial x}(x,y),\frac{\partial f}{\partial y}(x,y) \right)[/math]

 

So, if we want [math]\nabla f = F[/math], then f must satisfy:

 

[math]\frac{\partial f}{\partial x} = x \log y, \frac{\partial f}{\partial y} = y e^x[/math]

 

Now, if we simply integrate the first equation with respect to x, the second with respect to y we obtain the desired result. (Be careful about integration with respect to a single variable; instead of a constant of integration, you get a function which is dependent upon the variable you didn't integrate with respect to. e.g. in the first equation we get f(x,y) = ... + g(y)).