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Use polar coordinates to find the volume under $f(x,y)=1+(x^2+y^2)$ and over the disk $x^2+y^2\leq2$.

%%f(r,theta)=1+r^2%% and %%r^2<=2%%

%%int_0^(2pi)int_0^sqrt(2)(1+r^2)rdrd theta%% %%int_0^(2pi)int_0^sqrt(2)r+r^3drd theta%% %%int_0^(2pi)r^2/2+r^4/4 |_0^sqrt(2)%% %%int_0^(2pi)2d theta%% %%2 theta|_0^(2pi)=4pi%%

## Comments

%%f(r,theta)=1+r^2%% and %%r^2<=2%%

%%int_0^(2pi)int_0^sqrt(2)(1+r^2)rdrd theta%%

%%int_0^(2pi)int_0^sqrt(2)r+r^3drd theta%%

%%int_0^(2pi)r^2/2+r^4/4 |_0^sqrt(2)%%

%%int_0^(2pi)2d theta%%

%%2 theta|_0^(2pi)=4pi%%