We want to find the area of the region represented by the following inequality.
x^2+y^2 ≤ 72
In order to do so, let's first find the region given by it.
Shading half of the plane to show the solution set.
Let's start by plotting the boundary curve.
Drawing the Boundary Line
We can tell a lot of information about the boundary curve from the given inequality.
Exchanging the inequality symbols for equals signs gives us the boundary curve equation.
Observing the inequality symbols tells us whether the inequalities are strict.
Let's find each of these key pieces of information for the inequalities in the system.
Information
Inequality
Given Inequality
x^2+y^2≤72
Boundary Curve Equation
x^2+y^2=72
Solid or Dashed?
≥ ⇒ Solid
To graph the boundary curve equation, we will make a table of values.
x
x^2+y^2=72
y
- 6
( - 6)^2+y^2=72
± 6
- 3
( - 4)^2+y^2=72
± sqrt(63) ≈ ± 7.94
0
( 0)^2+y^2=72
± sqrt(72) ≈ ± 8.49
3
( 3)^2+y^2=72
± sqrt(63) ≈ ± 7.94
6
( 6)^2+y^2=72
± 6
We will now plot and connect the obtained points with a smooth curve.
As we can see, this curve is a circle with (0,0) as a center and a radius of sqrt(72) or about 8.49 units.
Shading the Solution Set
Before we can shade the solution set for the inequality, we need to determine on which side of the plane their solution sets lie. To do that, we will need a test point that does not lie on either boundary curve.
It looks like the point ( 0, 0) would be a good test point. We will substitute this point for x and y in the given inequality and simplify. If the substitution creates a true statement, we shade the same region as the test point. Otherwise, we shade the opposite region.
Since the statement is true, we will shade the region containing the test point.
Calculating the Area
We found that the region we want to find the area of is bounded by a circle, whose center is (0,0) and radius is sqrt(72) units. The area of a circle with radius r is calculated using the formula below.
A=π r^2
We know that the radius of the circle is sqrt(72). By substituting sqrt(72) for r in the formula, we can calculate A.
