Question

Evaluate the surface integral ∫∫ x^2 * z dσ
S
where the S is the part of the cylinder x^2 + z^2 =1 that is above the xy plane and between the planes y=0 and y=2. Can anyone help me?

Answer 1

Most of the questions i have to work with as examples use the double integral over σ and the dσ is dS... thats the first thing i am confused with...
the second thing is that i started and thought to write the equation of the cylinder in terms of x so i could let that equal to g(y,z) but since there is no y in either function I dont know how to integrate over dydz

Answer 2

\[x=\cos\phi,z=\sin\phi,d\sigma=d\phi dy\]
\[\iint\limits_Sx^2zd\sigma=\iint\limits_S\cos^2\phi\sin\phi d\phi dy=
\int\limits_0^\pi\cos^2\phi\sin\phi d\phi\cdot\int\limits_0^2dy=\]
\[=-2\int\limits_0^\pi\cos^2\phi\,d(\cos\phi)=-2\frac{cos^3\phi}{3}_{\phi=0}^{\phi=\pi}=\frac{4}{3}\]

Answer 3

the only problem with what you've written here is that you havent taken the g(x,y) and the partial derivatives of g with respect to x and y. the surface integral is supposed to look something like\[∫∫ f(x,y,z)*dS = ∫∫ f(x,y,z)*\sqrt{1+(∂x/∂y)^2+(∂x/∂z)^2}dA\]

Answer 4

Here is how I started.....
let x=\[\sqrt{1-z^2}\y
and ∂x/∂y = 0 and ∂x/∂z=\[∂x/∂z = -z/\sqrt{1-z^2}\]

so that my surface integral becomes
\[∫∫ x^2*z*dS\] =
\[[∫∫(1-z^2)*z*(1/\sqrt{1-z^2) }] = \]
\[∫∫(1-z^2)^0.5\]

I'm stuck here though because I would think the dA becomes dz*dy and I would integrate first with respect to z from 0 to 1 but then how to i integrate with respect to y if i dont have anything to find the range from

Answer 5

that first line that got cut off should say
∂x/∂z =

\[(-z)/(1-z^2)^0.5\]

Answer 6

Best way is
\[z=\sqrt{1-x^2}\,,\frac{\partial z}{\partial x}=\frac{-x}{\sqrt{1-x^2}}\,,\,\frac{\partial z}{\partial y}=0\]
\[\iint\limits_Sx^2zd\sigma=\iint x^2\sqrt{1-x^2}\sqrt{1+\frac{x^2}{1-x^2}}dxdy=
=\iint x^2dxdy=\]
\[=\int\limits_{-1}^{1}x^2dx\cdot\int\limits_{0}^{2}dy=\frac{2}{3}\cdot 2=\frac{4}{3}\]

Answer 7

so i was on the right track but i should have done the x^2 + z^2 = 1 in terms of z instead?! is there a reason?
also, for the range of y is just from the question since it said between the planes y=0 and y=2 right.? and for x is for when you set
\[z=\sqrt{(1-x^2)} \]
equal to 0 right and solve for x= +1 and -1

Answer 8

Find projection of surface S on xy plane, you will get rectangle (-1,0,0) , (1,0,0) , (1,2,0) , (-1,2,0).

Answer 9

thank you alot