I am studying calculus, and am wondering how to take a definite integral under the area of a circle. As a circle is $x^2 + y^2 = r^2$, how do you perform integration with a y in your formula?
My specific problem is $(x)^2 + (y-15)^2 =15^2 \{7.5<x<10\}$
Take your equation and solve for $y$.
$$x^2 + (y-15)^2 = 15^2$$ $$(y-15)^2 = 225 - x^2$$ $$y-15 = \pm \sqrt{225 - x^2}$$ $$y = \pm \sqrt{225 - x^2} + 15$$
We will want to use the negative square root because that gives us the bottom half of the circle. So, $$y = - \sqrt{225 - x^2} + 15.$$
Now you just integrate over the given domain.
$$\int_{7.5}^{10} - \sqrt{225 - x^2} + 15 dx$$