Q 39 - Calculus

Question

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

${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{r}f\left(r,\theta ,z\right)rdzdrd\theta$

Step-by-Step Solution

Answer

It represents the region below by the $r\theta -$  plane and bounded above by the cone $z=r$  on the circle with radius 3  centered at the origin.

Step 1:Given information

The given expression is ${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{r}f\left(r,\theta ,z\right)rdzdrd\theta$

Step 2:Simplificaion

Given 

${\int }_0^{2\mathrm{\pi }}{\int }_0^3{\int }_0^{r}f\left(r,\theta ,z\right)rdzdrd\theta$

From the limits of z

$\Rightarrow z=r$

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

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

From the limits of r

$r=3$

$\Rightarrow r^2=9$   (squaring both sides)

$\Rightarrow x^2+y^2=9$

This equation represents the equation of the circle with a radius of 3.

The limit of $\theta$ varies from $0 to 2\mathrm{\pi }$

Hence, the region below by the $r\theta -$ plane and bounded above by the cone

$z=r$ on the circle with radius 3 centered at the origin