Question 2.32: Sketch the region defined by (a) |x| <2 and |y| < 1 (......

(a) The region is a rectangle as shown in Figure 2.41. The boundary is not part of the region as strict inequalities were used to define it. The region |x| ≤ 2, |y| ≤  1 is the same as that in Figure 2.41 with the boundary included.

(b) \left|x^2+y^2\right| \leqslant 9 9 is equivalent to 9 is equivalent to. Note, however, that x^2+y^2 is never negative and so the region is given by 0 \leqslant x^2+y^2 \leqslant 9.

Let P(x, y) be a general point as shown in Figure 2.42. Then from Pythagoras’s theorem, the distance from P to the origin is \sqrt{x^2+y^2}. So,

Then,

This describes a disc, centre the origin, of radius 3 (see Figure 2.43).