Regions in Calculus

Inequalities can be shown as regions in the Cartesian plane. You can shade regions on a number plane that involve either linear or non-linear graphs. This means that we can have regions bounded by a circle or a parabola, or any of the other graphs you have drawn in this chapter.

Regions can be bounded or unbounded.

A bounded region means that the line or curve is included in the region.

Examples

1. Find the region defined by $y \ge x+2$.

First, sketch y = x + 2 as an unbroken line.

On one side of the line, y > x + 2 and on the other side, y < x + 2.

To find which side gives y > x + 2, test a point on one side of the line (not on the line).

For example, choose (-2, 1) and substitute into:

y > x + 2

1 > -2 + 2

1 > 0 (this is true)

This means that (-2, 1) lies in the region y > x + 2. The region is on the this (left or upper) side of the line. 2. Find the region defined by ${x}^{2} + {y}^{2} > 1$

The equation ${x}^{2}+{y}^{2}=1$ is a circle, radius 1 and centre (0, 0).

Draw ${x}^{2}+{y}^{2}=1$ as a broken line, since the region does not include the curve.

Choose a point inside the circle, say (0, 0) ${x}^{2}+{y}^{2}>1 \\ {0}^{2}+{0}^{2}>1 \\ 0 > 1$

(This is false)

So the region lies outside the circle. 3. Sketch the region $x\le 4,\quad y>-2\quad and\quad y\le { x }^{ 2 }$.