Q 40 - Calculus

Question

The iterated integrals use cylindrical coordinates. Describe the solids determined by the limits of integration. 

${\int }_0^{\mathrm{\pi }}{\int }_1^2{\int }_0^{r^2}f\left(r,\theta ,z\right)rdzdrd\theta$

Step-by-Step Solution

Answer

It represents  the region below by the $r\theta -$  plane and bounded above by the paraboloid $z=r^2$on the circles between radius  1 and 2

Step 1:Given information

The given expression is ${\int }_0^{\mathrm{\pi }}{\int }_1^2{\int }_0^{r^2}f\left(r,\theta ,z\right)rdzdrd\theta$

Step 2:Simplificaion

Given integral is defined,

${\int }_0^{\mathrm{\pi }}{\int }_1^2{\int }_0^{r^2}f\left(r,\theta ,z\right)rdzdrd\theta$

From the limits of z 

$z=r^2$

$\Rightarrow z=x^2+y^2$

This equation represents the equation of paraboloid centered at the origin.

From the limits of r. 

$r=1$

$r^2=1$

$\Rightarrow x^2+y^2=1$

And $r=2$,

$\Rightarrow r^2=2^2$ (squaring both sides)

$\Rightarrow x^2+y^2=2^2$

This equation represents the equation of circle varies from radius 1 o 2

Now,

The limits of $\theta$ varies from $\theta to \mathrm{\pi }$

Hence, the region below by the  $r\theta -$ plane and bounded above by the paraboloid $z=r^2$ on the circles between radius 1 and 2