Polar Co-ordinates

**adam_leeds** · April 13th 2011, 08:21 AM

Find the surface area of the circular paraboloid

$<span style=$ z = x^2+ y^2" alt="z = x^2+ y^2" />between

the xy-plane and the plane z = 4.

The limits have suddeny become for double integration

2pi 0
and 3 and 0
dr dtheta

I know

r = (x^2 + y^2)^0.5
x = r cos theta
y = r sin theta
and theta = inverse tan (x/y)

How do i get these limits? Thanks.

**adam_leeds** · April 13th 2011, 08:25 AM

$z = x^2 + y^ 2$ is the error

**FernandoRevilla** · April 13th 2011, 08:35 AM

The surface area is

$S=\displaystyle\iint_{x^2+y^2\leq 4}\sqrt{1+4x^2+4y^2}\;dxdy=$

$\displaystyle\int_{0}^{2\pi}d\theta\displaystyle\i nt_{0}^{2}\sqrt{1+4r^2}\;r\;dr=\ldots$

**adam_leeds** · April 13th 2011, 08:43 AM

Originally Posted by FernandoRevilla

The surface area is

$S=\displaystyle\iint_{x^2+y^2\leq 4}\sqrt{1+4x^2+4y^2}\;dxdy=$

$\displaystyle\int_{0}^{2\pi}d\theta\displaystyle\i nt_{0}^{2}\sqrt{1+4r^2}\;r\;dr=\ldots$

According to my book its 3 and 0.

How did you calculate the limits, i don't understand?

**FernandoRevilla** · April 13th 2011, 08:58 AM

Originally Posted by adam_leeds

According to my book its 3 and 0.

What is $3$ and $0$ ?. Could you rewrite the question?. Your first post says the plane $z=4$ , so the projection over the $xy$ plane is $x^2+y^4\leq 4$ .

**adam_leeds** · April 13th 2011, 09:10 AM

Originally Posted by FernandoRevilla

What is $3$ and $0$ ?. Could you rewrite the question?. Your first post says the plane $z=4$ , so the projection over the $xy$ plane is $x^2+y^4\leq 4$ .

Your 2nd integral you have got between 2 and 0 my book says its between 3 and 0

**adam_leeds** · April 13th 2011, 09:26 AM

Can you just tell me how to work it out please? Thanks.

**FernandoRevilla** · April 13th 2011, 10:01 AM

Originally Posted by adam_leeds

Your 2nd integral you have got between 2 and 0 my book says its between 3 and 0

Your book is wrong.

**FernandoRevilla** · April 13th 2011, 10:03 AM

Originally Posted by adam_leeds

Can you just tell me how to work it out please? Thanks.

Better fifty fifty.

First question:

Could you please write the projection of the given surface over the $xy$ plane ?

**adam_leeds** · April 13th 2011, 10:19 AM

Originally Posted by FernandoRevilla

Better fifty fifty.

First question:

Could you please write the projection of the given surface over the $xy$ plane ?

It just says

Find the surface area of the circular paraboloid z = x^2 + y^2 between the xy-plane and the plane z= 4

**FernandoRevilla** · April 13th 2011, 10:33 AM

Originally Posted by adam_leeds

It just says. Find the surface area of the circular paraboloid z = x^2 + y^2 between the xy-plane and the plane z= 4

Well, look at the answer #3 and tell me exactly what you don't undesrtand.

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