sketching regions of three variables

Question

If i had a solid region, V, such that $x^2+y^2+z^2\le9$, $x^2+y^2\le4$, $x\le0$ and $z\ge1$

what would be the easiest method to sketch this region? Can someone run me through steps as how to tackle these types of three variable regions?/ any helpful hints or tips?

i was unable to successfully sketch the above region, but i understand that $x^2+y^2+z^2\le9$ is a sphere with radius 3, $x^2+y^2\le4$ is a circle with radius 2, and that the region is limited to x values less than or equal to zero, while z is limited to values greater than or equal to 1.

From this, to describe in cylindrical coordinates i have deduced that

$0\le\phi\le??$ and $\pi/2\le\theta\le3\pi/2$

but i may be wrong as i have based this on the x and z restrictions. Also, how would i find my 'r' description? Is it to do with changing $x^2+y^2+z^2\le9$ and $x^2+y^2\le4$ into $r\le3$ and $r\le2$ and have $2\le r\le3$?

Thanks in advance :)!

Answer 1

• $x^2+y^2+z^2\le 9$ indicates the region inside an sphere with radii $3$ centered at the origin.

• $x^2+y^2\leq 4$ indicates the region inside a circular cylinder . Note that for this equation $z$ is a free parameter.

• $x\le 0$ forces us to select the region in which $x$ is negative. The same is for $z\ge 1$, i.e; we should choose the region where the $z$ component is greater or equal to $1$.

Now consider all above at the same time by drawing the sphere and the cylinder in a $xyz$ coordinate system.

• As you can see blow the ranges for $r$ and $\theta$ are as you achieved, but if in Cylindrical coordinates we don't have $\phi$. Indeed, we have $z$ instead. The range for it is $1\leq z\leq\sqrt{9-r^2}$