# Changing the order of integration

1. Mar 1, 2009

### a1010711

1. The problem statement, all variables and given/known data

Change the order of integration in the following double integral

integral from o to a, integral from 0 to sqrt(2ay-y^2) f(x y) dx dy

so i can see its a semi circle with center at (0,a)
x= sqrt(2ay-y^2) can be expanded by squaring both sides. then completing the square x^2+(y-a)^2=a^2 which is a cirle of radius 'a', center at (x,0)

then im not sure what to do.. how do i integrate with respect to y first,,then x? how do i set this up

2. Mar 1, 2009

### Staff: Mentor

The center of the circle is at (0, a), not (x, 0).
In the original iterated integral, x ranges from 0 to sqrt(2ay - y^2), or from the y-axis to the right half of the circle. y ranges from 0 to a, so the region of integration is the lower right-hand quarter of this circle.

When you change the order of integration, the y values will need to range from the lower part of the circle up to the line y = a, and the x values will need to range from the line x = 0 to the right hand edge of the circle.