Find the centroid of the region bounded by the graphs of the given equations. $y=|x| \sqrt{4-x^{2}}, \quad y=0, \quad x=-2, \quad x=2$
A) $\left(0, \frac{4}{5}\right)$
B) $\left(\frac{5}{4}, 0\right)$
C) $\left(\frac{4}{5}, 0\right)$
D) $\left(0, \frac{5}{4}\right)$

Question

Quizplus · Accepted Answer

The region is symmetric about the y-axis, so the x-coordinate of the centroid is 0. The area of the region is $$A=\int_{-2}^{2} |x| \sqrt{4-x^{2}} \ dx = 4\int_{0}^{2} x \sqrt{4-x^{2}} \ dx$$$$=4\left[-\frac{1}{3}(4-x^{2})^{3/2}\right]_{0}^{2}=\frac{32}{3}$$So, the y-coordinate of the centroid is $$\overline{y}=\frac{1}{A}\int_{-2}^{2} y \ dA = \frac{1}{\frac{32}{3}}\int_{-2}^{2} |x| \sqrt{4-x^{2}} \ y \ dx$$$$=\frac{3}{32}\int_{-2}^{2} |x| (4-x^{2}) \ dx = \frac{3}{32}\left[4\int_{0}^{2} x(4-x^{2}) \ dx\right]$$$$=\frac{3}{32}\left[4\left(-\frac{4}{3}x^{3}+\frac{x^{5}}{5}\right)\right]_{0}^{2}=\frac{4}{5}$$Therefore, the centroid is $(0, \frac{4}{5})$.

Find the Centroid of the Region Bounded by the Graphs y=∣x∣4−x2,y=0,x=−2,x=2y=|x| \sqrt{4-x^{2}}, \quad y=0, \quad x=-2, \quad x=2y=∣x∣4−x2​,y=0,x=−2,x=2

Find the Centroid of the Region Bounded by the Graphs $y=|x| \sqrt{4-x^{2}}, \quad y=0, \quad x=-2, \quad x=2$