Double Integration in Polar and its Applications

Well for sec^3 x, you use d/dx tan x = sec^2 x, so let dv = sec^2 x dx
Anyway I'm not clear on how you figured out the limits, I'd go about it something like

 Spoiler

y = sqrt(2x-x^2)
y^2 = 2x-x^2
x^2+y^2 = 2x
r^2 = 2rcos(t)
r=2cos(t)

which doesn't match up with what you got. That has the integral

 Spoiler

#1 really should be easier with x,y coordinates - putting it in as written in Wolfram Alpha gives 54, it's just a matter of doing the integral correctly I think. (antiderivative of x+y with respect to y is xy+1/2y^2 | y=x/2 to 3-x, which should give you a polynomial of x)

#2 I would also not do with polar coordinates unless you have to.