Use polar coordinates to combine the sum into a double integral, then integrate.
integral #1 outer limit, x goes from 1/sqrt(2) to 1, y goes from sqrt of 1-x^2 to x
integrand xy dy dx
add to integral #2 outer limits, x geos from 1 to sqrt2, and y goes from 0 to x
integrand is xy dy dx
add to integral #3 outer limits, x goes from sqrt2 to 2, and y goes from 0 to sqrt(4-x^2)
integrand is again xy dy dx
I graphed the three in xy plane to see their picture. I am just not seeing the polar on this one. I was thinking
V= outer integral 0 to pi/4, inner sqrt 2 to 2 r^2cos(theta)sin(theta) r dr dtheta.