Find the area of the parts formed from the parabola $y=x^2-5$ separating the circle $x^2 + y^2 \leq 5^2$?

Question

I know the standard method when we are given equations, but what do we do when we have inequalities?

I know the radius of the circle is $5$.

I drew both of the graphs together but I don't know what to do after.

Answer 1

For checking an inequality of an equation it will be a one of the regions the curve divides the plane into. You can check this by substituting a point. eg. Here the region will be either inside or outside the circle. To check you know (0,0) is a point inside the circle and it satisfies the inequation. So actually all points inside circle will satisfy the same. The region is interior of the circle. So you have to find the area cut out by the parabola from the interior of the circle. The area can be found by evaluating: $2\int_0^3 (\sqrt {25-x^2} - (x^2-5))dx$