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