Attached are a question and the matlab code. The code gives the total surface area as 639.4828, but the figure is not showing up. Also, the...

i, j, k are unit vectors.

r(u,v)=(b+a*cos(u))*cos(v) i + (b+a*cos(u))*sin(v) j + a*sin(u) k

0<=u,v<=2pi and a=1/4 and b=1

The surface area is:

A=double integral of magnitude of (r_u x r_v) dA

r_u is the partial derivative of r(u,v) with respect to u and r_v with respect to v:

r_u(u,v)= -a*sin(u)*cos(v) i -a*sin(u)*sin(v) j +a*cos(u) k

r_v(u,v)= -(b+a*cos(u))*sin(v) i +(b+a*cos(u))*cos(v) j

Compute the cross product:

r_u x r_v = -a*cos(u)*(b+a*cos(u))*cos(v) i - a*cos(u)*(b+a*cos(u))*sin(v)...

i, j, k are unit vectors.

r(u,v)=(b+a*cos(u))*cos(v) i + (b+a*cos(u))*sin(v) j + a*sin(u) k

0<=u,v<=2pi and a=1/4 and b=1

The surface area is:

A=double integral of magnitude of (r_u x r_v) dA

r_u is the partial derivative of r(u,v) with respect to u and r_v with respect to v:

r_u(u,v)= -a*sin(u)*cos(v) i -a*sin(u)*sin(v) j +a*cos(u) k

r_v(u,v)= -(b+a*cos(u))*sin(v) i +(b+a*cos(u))*cos(v) j

Compute the cross product:

r_u x r_v = -a*cos(u)*(b+a*cos(u))*cos(v) i - a*cos(u)*(b+a*cos(u))*sin(v) j - a*(b+a*cos(u))*sin(u) k

Compute the magnitude:

magnitude(r_u x r_v)= sqrt(a^4*(cos(u))^2+2*a^3*b*cos(u)+a^2*b^2)= a*(b+a*cos(u))

Lastly, evaluate the integral:

integral(integral(a*(b+a*cos(u)),v,0,2*pi),u,0,2*pi) = 4*pi^2 * a * b

With a=1/4 and b=1 the surface area is pi^2.

The Matlab code for the plot is:

clear all
syms u v
a=1/4; b=1;
x=(b+a*cos(u))*cos(v);
y=(b+a*cos(u))*sin(v);
z=a*sin(u);
ezsurf(x,y,z,[0,2*pi,0,2*pi])