What does this set look like?

Question

I have to find the volume integral of the set

$$M=\{x \in \mathbb{R}^2 \mid 1 \leq x_1^2 + x_2^2, |x_1| \leq 1, |x_2| \leq 1\}$$

But I can't figure how it looks like, so I can't set the bounds of the integral.

Does anyone know how it looks like in the $x_1$-$x_2$ plane?

The absolute value bounds mean you're inside a square. The other bound means you're outside the unit circle. — Paul, Jan 16 '16 at 20:27
@Paul of course... Didn't think of a square but a second circle. Stupid. Thank you — JavaForStarters, Jan 16 '16 at 20:29

score 3 · Accepted Answer · answered Jan 16 '16 at 20:29

3

Hint: The set of points $A=\{(x_1,x_2)\in \mathbb{R}^2:|x_1|\leq 1,|x_2|\leq 1\}$ is a square (with interior) with vertices $(1,1)$, $(1,-1)$, $(-1,1)$, and $(-1,-1)$. The set of points $B=\{(x_1,x_2)\in \mathbb{R}^2:x_1^2+x_2^2<1\}$ is the unit disk with boundary removed. Can you see that $M=A\setminus B$? Using this you can compute the area of $M$ by subtracting the area of $B$ from the area of $A$.

answered Jan 16 '16 at 20:29

TomGrubb

12,909

1

Giving an area of $4-\pi.$............. – DanielWainfleet Jan 16 '16 at 23:17

What does this set look like?

1 Answers1