Thread: Double integrals into polar coordinates

1. Double integrals into polar coordinates

Use the double integral in polar coordinates to find the volume of the solid bounded by the graphs of the given equuations :

z= x^2+y^2+3 , z=0 , x^2+y^2=1

2. Originally Posted by cchyfly
Use the double integral in polar coordinates to find the volume of the solid bounded by the graphs of the given equuations :

z= x^2+y^2+3 , z=0 , x^2+y^2=1
$\iint_Dx^2+y^2+3dA=\int_{0}^{2\pi}\int_{0}^{1}(r^2 +3)rdrd\theta$

$\int_{0}^{2\pi}\left( \frac{1}{4}r^4+\frac{3}{2}r^2\right)|_{0}^{1}d\the ta=\frac{7}{4}\int_{0}^{2\pi}d\theta=\frac{7\pi}{2 }$

3. Hello TheEmptySet,

I have a question to this (I'm trying to understand these problems) :
Where did we use the fact that it was bounded by z=0 ?

4. Originally Posted by Moo
Hello TheEmptySet,

I have a question to this (I'm trying to understand these problems) :
Where did we use the fact that it was bounded by z=0 ?
The integrand $x^2+y^2+3$
is the height above the xy plane. As we integrate over the area in the xy plane we get the volume between the surface and the plane z=0.

The 2d equivelent is finding the area under the curve $y=x^2$ from x=2 to x=4. We use the fact that it is bounded below by y=0 to get the area between the curve and the x axis.

I hope this Helps.

Brett

5. Originally Posted by TheEmptySet
The integrand $x^2+y^2+3$
is the height above the xy plane. As we integrate over the area in the xy plane we get the volume between the surface and the plane z=0.

The 2d equivelent is finding the area under the curve $y=x^2$ from x=2 to x=4. We use the fact that it is bounded below by y=0 to get the area between the curve and the x axis.

I hope this Helps.

Brett
Ouh, thanks !
It's (quite ) ok for the first part, but I don't see where x² comes from in the second part ? And how do you get the boundaries 2 & 4 ?

6. Originally Posted by Moo
Ouh, thanks !
It's (quite ) ok for the first part, but I don't see where x² comes from in the second part ? And how do you get the boundaries 2 & 4 ?

O just made that up for the sake of an example. It is not related to the prevoius problem. Sorry.

Brett