Calculus - Lamina / Integral Bounds?

Question

A lamina has the shape of a closed region bounded by the graphs of x^2 + y^2 = 4 and x + y = 2, and has the density
p(x,y) = x*y. Write the iterated integral for the moment of inertia about the y-axis. Then evaluate the integral.

I am unsure about how to write the bounds for this, I suppose you could use polar form and write r=2, but I am not too sure how to go from there if that is the case in terms of bounds etc. to set it up properly.

Answer 1

I would stick with xy-coordinates.

If you haven't done so already, draw a picture of the region. The lamina is the part of the circle in the first quadrant that is cut off by the line.

If you want to do x-integration first:

for the x limits, x goes from the line to the circle.

Line: x = y - 2 (lower limit of integration)
Circle: x = √(4 - y²) (upper limit of integration)

Then y goes from 0 to 2 (the intersection points of the circle and the line).

If you want to do y-integration first, because of the symmetric way in which x and y appear in the equations of the boundary, the limits are very similar:

Line: y = x - 2 (lower limit of integration)
Circle: y = √(4 - x²) (upper limit of integration)

Then x goes from 0 to 2.