Section 15.4 (6th edition): Exercise 8: Evaluate over the region between x^2+y^2=1 and x^2+y^2=4 on the left of the y-axis.

f(x,y) := x+y;

integrate(integrate(f(r*cos(s),r*sin(s))*r,r,1,2),s,%pi/2,3*%pi/2);

Exercise 12: Evaluate over the region enclosed by the circle of radius 5 in the first quadrant. Here the order of iterations does matter

f(x,y) := y*exp(x);

integrate(integrate(f(r*cos(s),r*sin(s))*r,s,0,%pi/2),r,0,5), expand;

If you integrate it with respect to r first, even maxima fails to find the simplest
answer.

integrate(integrate(f(r*cos(s),r*sin(s))*r,r,0,5),s,0,%pi/2), expand;

Exercise 30: Evaluate integrate(integrate(x^2*y,x,-sqrt(a^2-y^2),0),y,0,a)

f(x,y) := x^2*y;

integrate(integrate(f(r*cos(s),r*sin(s))*r,r,0,a),s,%pi/2,%pi);

Exercise 32: Evaluate integrate(integrate(sqrt(x^2+y^2),y,0,sqrt(2x-x^2)),x,0,2). Review Example 4 first. Then try to find the region in terms of polar coordinate.

integrate(integrate(r^2,r,0,2*cos(s)),s,0,%pi/2);