Section 15.2: Example 1: Double integral over type I region bounded by

g(x) := 2*x^2;

h(x) := 1+x^2;

Identify the interval for x from range[1] to range[2] in the region of type I

solve(g(x)=h(x));

range: map(rhs, solve(g(x)=h(x)));

wxplot2d([g(x),h(x)],[x,range[1],range[2]],[legend,"g(x)","h(x)"]);

integrate(integrate(x+2*y,y,g(x),h(x)),x,range[1],range[2]);

Example 3: Double integral over type II region bounded by

h: rhs(solve(y=x-1,x)[1]);

g: rhs(solve(y^2=2*x+6,x)[1]);

Identify the interval for y from range[1] to range[2] in the region of type II

solve(g=h);

range: map(rhs, solve(g=h));

wxplot2d([g,h],[y,range[1],range[2]],[legend,"g","h"]);

integrate(integrate(x*y,x,g,h),y,range[1],range[2]);

Example 4: Volumn of tetrahedron bounded by the surface

f: rhs(solve(x+2*y+z=2,z)[1]);

The volume over type I region bounded by

g: rhs(solve(x=2*y,y)[1]);

h: rhs(solve(x+2*y=2,y)[1]);

Identify the interval for x from x=0 to range[2] in the region of type I

solve(g=h);

range: [0,rhs(solve(g=h)[1])];

wxplot2d([g,h],[x,range[1],range[2]],[legend,"g","h"]);

integrate(integrate(f,y,g,h),x,range[1],range[2]);