1. ## Surface Integrals

I have to evaluate the given surface integrals $\iint s {z}^{2 } d\sigma$ where S is the portion of the cone $z = \sqrt{{x}^{2 } +{y}^{2 } }$ between the planes z = 1 and z = 2

I dont know where to start

I have to evaluate the given surface integrals $\iint s {z}^{2 } d\sigma$ where S is the portion of the cone $z = \sqrt{{x}^{2 } +{y}^{2 } }$ between the planes z = 1 and z = 2

I dont know where to start
The top and bottom are circles or radius 2 and 1 respectivily and can be parametrized using cylindrical polar coordinates.

Note that on the side the azimuthal angle is fixed at

$\theta=\frac{\pi}{4}$

So the surface can be parameterized as follows

$x=r\sin\left(\frac{\pi}{4} \right)\cos(\phi) \quad y=r\sin\left(\frac{\pi}{4} \right)\sin(\phi) \quad z=r\cos\left(\frac{\pi}{4} \right)$

So as a vector we get

$\mathbf{r}(r,\phi)=[r\sin\left(\frac{\pi}{4} \right)\cos(\phi)]\mathbf{i}+[r\sin\left(\frac{\pi}{4} \right)\sin(\phi)]\mathbf{j}+[r\cos\left(\frac{\pi}{4} \right)]\mathbf{k}$

Can you finish from here?

3. I'll have a go, whats the formula for getting theta?

Is the next step F.r' ?

4. You can use the standard method

$\displaystyle\iint_S F(x,y,z)dS=\displaystyle\iint_DF(x,y,f(x,y))\sqrt{ 1+\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial y}}\;dxdy$

In this case

$F(x,y,z)=z^2,\;z=f(x,y)=\sqrt{x^2+y^2},\;D\equiv 1\leq x^2+y^2 \leq 4$

Edited: Sorry, I didn't see the TheEmptySet's post.

I'll have a go, whats the formula for getting theta?

Is the next step F.r' ?
You have the cone

$z=\sqrt{x^2+y^2}=r$

From spherical coordinates we know that

$z=r\cos(\theta) \iff \theta =\cos^{-1}\left( \frac{z}{r}\right)$

but from the equation of the cone we know that

$z=r$

So we get

$\theta =\cos^{-1}\left( \frac{r}{r}\right) =\cos^{-1}\left( 1\right)=\frac{\pi}{4}$

If we have the vector form of a surface its surface area is given by

$\iint \bigg|\frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \bigg| dr d\theta$

6. Originally Posted by FernandoRevilla
You can use the standard method

$\displaystyle\iint_S F(x,y,z)dS=\displaystyle\iint_DF(x,y,f(x,y))\sqrt{ 1+\dfrac{\partial f}{\partial x}+\dfrac{\partial f}{\partial y}}\;dxdy$

In this case

$F(x,y,z)=z^2,\;z=f(x,y)=\sqrt{x^2+y^2},\;D\equiv 1\leq x^2+y^2 \leq 4$

Edited: Sorry, I didn't see the TheEmptySet's post.
This method seems more familiar to me,

$F(x,y,f(x,y)) = {x}^{ 2} + {y}^{ 2}$ , is that correct?

7. Yes