Section 15.5: Example 1: Surface area over R = {(x,y): 0<x<1, 0<y<x}

draw3d(xlabel="x-axis",ylabel="y-axis",surface_hide=true,color=red,explicit(x^2+2*y,x,0,1,y,0,1),color=blue,implicit(y=x,x,0,1,y,0,1,z,0,3));

f: x^2 + 2*y;

A: sqrt(1 + diff(f,x)^2 + diff(f,y)^2);

'integrate('integrate(A,y,0,x),x,0,1);

integrate(integrate(A,y,0,x),x,0,1);

Example 2: Surface area over R = {(r*cos(theta),r*sin(theta)): 0<r<3, 0<theta<2*%pi}

f: x^2+y^2;

A: sqrt(1 + diff(f,x)^2 + diff(f,y)^2);

B: A,x=r*cos(theta),y=r*sin(theta);

'integrate('integrate(B*r,r,0,3),theta,0,2*%pi);

'integrate('integrate(trigsimp(B*r),r,0,3),theta,0,2*%pi);

integrate(integrate(B*r,r,0,3),theta,0,2*%pi);