restart;
f := r->arctan(r);
x := r*cos(theta);
y := r*sin(theta);
z := f(r);
plot3d([x, y, z], r = 0 .. 2*Pi, theta = 0 .. 2*Pi);

restart;
V := int(Pi*tan(y)^2, y = 0 .. arctan(2*Pi));
W := int(2*Pi*(arctan(2*Pi)-arctan(x))*x, x = 0 .. 2*Pi);

restart;
f := r->arctan(sqrt(r));
x := r*cos(theta);
y := r*sin(theta);
z := f(r);
plot3d([x, y, z], r = 0 .. 2*Pi, theta = 0 .. 2*Pi);

restart;
int(Pi*tan(y)^4, y = 0 .. arctan(sqrt(2*Pi)));
int(2*Pi*(arctan(sqrt(2*Pi))-arctan(sqrt(x)))*x, x = 0 .. 2*Pi);

restart;
simplify(arctan(sin(x)));
plot(7+arctan(sin(x)), x = 0 .. 2*Pi);

restart;
f := r->arctan(sin(r));
x := r*cos(theta);
y := r*sin(theta);
z := f(r);
plot3d([x, y, z], r = 0 .. 3*Pi, theta = 0 .. 2*Pi);

restart;
V := int(2*Pi*(7+arctan(sin(x)))*x, x = 0 .. 3*Pi);
evalf(V);
Digits := 40;
evalf(V);