Question

Find the centroid of the area bounded by

x² + y² = 25, x + y = 5

Answer 1

Solution.

$S=\frac{1}{4}πr^2-\frac{1}{2}•5•5=\newline \frac{25}{4}π-\frac{25}{2}=7.$

$x_C=\frac{1}{S}\int_0^5 (x(\sqrt{25-x^2}-(5-x))dx=\newline =\frac{1}{7}\int_0^5 (x\sqrt{25-x^2}-5x+x^2))dx=\newline =\frac{1}{7}•\frac{125}{6}=\frac{125}{42}.$

$y_C=\frac{1}{2S}\int_0^5(25-x^2-(5-x)^2)dx=\newline \frac{1}{14}\int_0^5(-2x^2+10x)dx=\newline \frac{1}{14}(-\frac{2}{3}x^3+5x^2)|_0^5=\newline \frac{1}{14}•\frac{125}{3}=\frac{125}{42}.$

So, centroid $(x_C,y_C)=(\frac{125}{42},\frac{125}{42}).$

Answer. $(\frac{125}{42},\frac{125}{42}).$