Express as an area integral in polar coordinates the area of the surface

z= y^2 +2xy -x^2 +2

lying over the annulus defined by 8/3 ≤ x^2 + y^2 ≤ 1.

Hence evaluate this area in terms of π.

Thanks

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- Sep 18th 2010, 03:13 AMacu04385Area integral in polar co-ords of a surface
Express as an area integral in polar coordinates the area of the surface

z= y^2 +2xy -x^2 +2

lying over the annulus defined by 8/3 ≤ x^2 + y^2 ≤ 1.

Hence evaluate this area in terms of π.

Thanks

- Sep 18th 2010, 03:48 AMHallsofIvy
Have you made any attempt at this at all?

The area you are to integrate over, in polar coordinates, is just $\displaystyle 8/3\le r^2\le 1$ which, since r must be positive is exactly $\displaystyle \sqrt{8/3}\le r\le 1$. That makes the limits of integration easy. And don't forget that the "differential of area" in polar cordinates is $\displaystyle rdrd\theta$. Write $\displaystyle z= y^2+ 2xy- x^2+ 2$ in polar coordinates and integrate!

Ouch! I just realized this is a**surface**integral, not a volume integral. But the "polar coordinates" part is the same. To get the surface area integrand you can think of the surface $\displaystyle z= y^2+ 2xy- x^2+ 2$ as given by the vector function $\displaystyle \vec{r}= x\vec{i}+ y\vec{j}+ z\vec{k}= x\vec{i}+ y\vec{j}+ (y^2- 2xy- x^2+ 2)\vec{k}$ with the x and y coordinates as parameters. The derivative vectors $\displaystyle \vec{r}_x= \vec{i}- (2y+ 2x)\vec{k}$ and $\displaystyle \vec{r}_y= \vec{j}+ (2y- 2x)\vec{k}$ are tangent to the surface and their cross product,

$\displaystyle \left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 1 & 0 & -(2y+2x) \\ 0 & 1 & 2y- 2x\end{array}\right|= -(2y+ 2x)\vec{i}- (2y+ 2x)\vec{j}+ \vec{k}$

is perpendicular to the surface and gives the "vector differential of surface area". Its length, $\displaystyle \sqrt{1+ 8x^2+ 8y^2}$ gives the "scalar differential of surface area", $\displaystyle \sqrt{1+ 8x^2+ 8y^2}dx dy$. I think you can see that will be much easier to integrate in polar coordinates!