We are asked to find the area of the graph of the solution region of the given equation. Looking at it, it seems that the inequality describes a circle. To make sure of that, let's first find the equation of the boundary curve.
Boundary Curve
In order to find the equation of the boundary curve from the inequality, we need to change the inequality sign into the equality sign.
x^2+y^2 ≤ 49
⇕
x^2 + y^2 = 49
Now, since we suspect that the boundary curve might be a circle, let's recall its standard equation.
(x- h)^2+(y- k)^2= r^2
Here, the center is the point ( h, k) and the radius is r. We will try to rewrite the given equation to match this form, and then we can identify the center and the radius. In this case, we will need to subtract 0 from the variable terms twice and ensure that the constant term is a square.
As we can see, the equation for the boundary curve matches the standard equation of a circle. The center of this circle is the point ( 0, 0), and its radius is 7. Let's graph it!
Solution Region
Now that we have the boundary curve, we need to make sure whether our inequality represents the outside or the inside of the region enclosed by the curve. In order to do so, let's test the inequality with a point outside the curve, for example, point (0,0).
Since the substitution gave us a true inequality, our inequality represents the region containing point (0,0). Let's shade it.
Area
Now that we have graphed our inequality, we can move to finding the are of the solution region. Since it is a circle, we can use the formula for the area of a circle to do so.
A = π r^2
We already found out that r = 7, so we can substitute it into formula above and solve for the area.
