Q 35 - Calculus

Question

Describe the graphs of the equations  and provide alternative equations in the specified coordinate systems 

Change $\rho =2$ to the rectangular and cylindrical systems

Step-by-Step Solution

Answer

The equation in rectangular coordinates is $x^2+y^2+z^2=4$

The equaion in cylindrical coordinates is $r^2+z^2=4$

Step 1:Given information

The given expression is $\rho =2$

Step 2:Simplification

To convert $\rho =2$ into rectangular coordinates

In rectangular coordinates we have

$x^2+y^2+z^2={\rho }^2$

Since $\rho =2$

$\Rightarrow x^2+y^2+z^2=4$

So, it represents a sphere with center at origin

Step 3:Simplification

To convert $\rho =2$into cylindrical coordinates we have

$x=r\cos \theta ,y=r\sin \theta ,z=z$

substituting in the above equation

width="198" height="22" style="max-width: none;" $\Rightarrow r^2{\cos }^2\theta +r^2{\sin }^2\theta +z^2=4$

width="195" height="22" style="max-width: none;" $\Rightarrow r^2\left({\cos }^2\theta +{\sin }^2\theta \right)+z^2=4$

$\Rightarrow r^2+z^2=4$

Therefore,the equaion in cylindrical coordinates is $r^2+z^2=4$