I apologize that this is a basic question.
I have a circle with radius $2.5 \sqrt{2}$ (the center is the origin). Consider the arc of the circle between $x=0$ and $x=1$. I want to know if the area between the arc and the line $y=4$ is greater than $1/2$.
The formula for a circle is $x^2 + y^2 = r^2$. I know $r$ so it feels like I should do an integral between $0$ and $1$ but I don't see how.
You should integrate the difference between $y=4$ and $y = \sqrt{r^2-x^2} \;$ on the range $x=0$ to $x=1$.
That's the area you want.