Integrating f(x,y) with Known p^2

[Niles]

Homework Statement

Hi

Say I have the function f(x,y) = xy, and I want to integrate f(x,y) from (0,0) to some (p_x, p_y), where I know $p^2 = p_x^2+p_y^2$. What I have done is to write

$$p_x ^2 + p_y ^2 = p^2$$

so the limits for p_x run from $\pm \sqrt {p^2 - p_y ^2 }$. Now, how about the limits for p_y? Note, that I only know the value of p, not p_x or p_y.

 

[Mark44]

Are you doing something like this?
$$\int \int_D xy dA$$

How would you describe the region D over which integration is taking place?

 

[Niles]

I am doing

$$\int \int_l xy dxdy$$

The line l is the diagonal going from (0,0) to some p, which is known (but whose components are now known). My issue is that I am not sure how to express the second variable (as in my OP).

 

[Dick]

I am doing

$$\int \int_l xy dxdy$$

The line l is the diagonal going from (0,0) to some p, which is known (but whose components are now known). My issue is that I am not sure how to express the second variable (as in my OP).

If you are trying integrate f(x,y)=xy along a curve 'l', then it's a one dimensional integration. You don't want to integrate dx*dy. Write the contour 'l' as a function of a single variable, say 't'. Then integrate dt.