Problem |
> y^2+z^2 = 9;
2 2 y + z = 9
that lies inside the cylinder |
> x^2+z^2 = 9;
2 2 x + z = 9
|
Solution
The part of the surface S above the xy plane is characterized by
|
> with(linalg): i := vector([1,0,0]): j:=vector([0,1,0]): k:=vector([0,0,1]):
> r = x*i + y*j + sqrt(9-y^2)*k;
2 1/2 r = x i + y j + (9 - y ) k
The element of surface area (sans dxdy) is given by |
> z := sqrt(9-y^2): dS := sqrt(1+diff(z,x)^2+diff(z,y)^2);
/ 2 \1/2 | y | dS := |1 + ------| | 2| \ 9 - y /
the total area is then, |
> Answer := 2*Int(Int(dS,x=-abs(y)..abs(y)),y=-3..3) =
> 2*int(int(dS,x=-abs(y)..abs(y)),y=-3..3);
Answer := 3 abs(y) 3 / / / 2 \1/2 / | | | y | | / 1 \1/2 2 | | |1 + ------| dx dy = 2 | 6 |- --------| abs(y) dy | | | 2| | | 2| / / \ 9 - y / / \ - 9 + y / -3 - abs(y) -3
and this evaluates to: |
> evalf(rhs("));
72.00000000