Examples, Section 15.4
MAT329, Fall, 2015
Andy Long

#5, p. 1026:

Clear[a,b,f,r,th]
f[x_,y_]:=1 (* hmmm.... what's that all about?!:*)
Integrate[f[r Cos[th],r Sin[th]]*r,{th,Pi/4,3Pi/4},{r,1,2}]

#10, p. 1026: here’s a pretty classic one: just implement the transformation as in the text, and get your limits right:

Clear[a,b,f,r,th]
f[x_,y_]:=y^2/(x^2+y^2)
Integrate[f[r Cos[th],r Sin[th]]*r,{th,0,2Pi},{r,a,b}]

#14, p. 1026: getting the limits right here (or thinking about it correctly) is the tricky part:

f[r_,th_]:=r Cos[th]
Integrate[f[r,th]*r,{th,0,Pi/2},{r,0,2}]-Integrate[f[r,th]*r,{th,0,Pi/2},{r,0,2Cos[th]}]
(* Amit had a good idea: *)
Integrate[f[r,th]*r,{th,0,Pi/2},{r,0,2}]-Integrate[(f[r,th]+1)*r,{th,0,Pi},{r,0,1}]

#36, p. 1027: crazy sprinkler!

w[r_,th_]:=E^(-r)
Integrate[r w[r,th],{r,0,R},{th,0,2Pi}]
radius=2 (* supposed to be 100, but what happens to the exponential out there? *)
ParametricPlot3D[{u Cos[v],u Sin[v],w[u,v]},{u,0,radius},{v,0,2Pi},PlotStyle->Directive[Green,Opacity[0.8],Specularity[White,20]],PlotRange->All]

An illustration that the area of a “polar rectangle” tends to the rectangular rdrdtheta as dr and dtheta tend to zero....

center=1.0/Sqrt[2];
Manipulate[
Plot3D[1,{x,center-c,center+c},{y,center-c,center+c},RegionFunction->Function[{x,y,z},1<x^2+y^2<1+c&&Pi/4-c/2<ArcTan[y/x]<Pi/4+c/2]],
{c,1,.000001}]