Finding the area under a [translated] circle using definite integrals

Question

I'm trying to find the area of a semi-circle using definite integrals, but I'm not understanding the results.

www.desmos.com/calculator/1ucgmybjxx

It looks fine when $k=0$, but when translating the figure down, the area turns negative before the top is even under the $x$-axis.

What's going on?

Answer 1

Create $$g(x)=-\sqrt{r^2-(x-r)^2}+k$$ This is the lower half of the circle. Calculate $$\int_0^{2r}(f(x)-g(x))dx$$ If you calculate only integral of $f$ you calculate the area of the half of the disk plus the area between the horizontal diameter and the $x$ axis. In my case, this area will cancel

Answer 2

You're essentially evaluating the area of the semi-circle over the $x$-axis, minus the area below the $x$-axis. If there's more area below than above, your integral will be negative.

Recall that when evaluating an integral as the limit of the area of rectangles, any rectangle below the $x$-axis counts as negative.