"Surface area"

clear
M1 = ((0,0,0),(0,0,-1),(0,1,0))
M2 = ((0,0,1),(0,0,0),(-1,0,0))
M3 = ((0,-1,0),(1,0,0),(0,0,0))
M = (M1,M2,M3)
cross(u,v) = dot(u,M,v)
z=2
S = (x,y,z)
a = abs(cross(d(S,x),d(S,y)))
T = defint(a,y,-sqrt(1 - x^2),sqrt(1 - x^2),x,-1,1)
check(T = pi)

clear
M1 = ((0,0,0),(0,0,-1),(0,1,0))
M2 = ((0,0,1),(0,0,0),(-1,0,0))
M3 = ((0,-1,0),(1,0,0),(0,0,0))
M = (M1,M2,M3)
cross(u,v) = dot(u,M,v)
z = x^2 + 2y
S = (x,y,z)
a = abs(cross(d(S,x),d(S,y)))
T = defint(a,x,0,1,y,0,1)
check(T = 5/8 log(5) + 3/2)

clear
M1 = ((0,0,0),(0,0,-1),(0,1,0))
M2 = ((0,0,1),(0,0,0),(-1,0,0))
M3 = ((0,-1,0),(1,0,0),(0,0,0))
M = (M1,M2,M3)
cross(u,v) = dot(u,M,v)
x = u cos(v)
y = u sin(v)
z = v
S = (x,y,z)
a = circexp(abs(cross(d(S,u),d(S,v))))
T = defint(a,u,0,1,v,0,3pi)
check(T = 3/2 pi log(1 + sqrt(2)) + 3 pi / sqrt(2))
