Evaluating a semicircle integral using signed area

The equation $y = -\sqrt{4-x^2}$ describes the lower half of the circle $x^2 + y^2 = 4$, which has radius $r = 2$. This lower half is a semicircle lying entirely below the x-axis on $[-2,2]$.

The area of a full circle of radius 2 is $\pi r^2 = 4\pi$, so the area of the semicircular region is $2\pi$. Since the region lies below the x-axis, it contributes negatively to the signed area, so $\int_{-2}^{2} -\sqrt{4-x^2}\,dx = -2\pi$.

The integral value is negative. Since the semicircular region in part a lies entirely below the x-axis, it contributes negative signed area to the definite integral. This matches the computed result of negative two pi, confirming the sign follows directly from the region's position relative to the axis.