Answer to Question #186282 in Differential Geometry | Topology for Jethro

Question #186282

Find the centroid of the area bounded by

x²+ y² = 25, x + y = 5

Expert's answer

Solution.

"S=\\frac{1}{4}\u03c0r^2-\\frac{1}{2}\u20225\u20225=\\newline\n\\frac{25}{4}\u03c0-\\frac{25}{2}=7."

"x_C=\\frac{1}{S}\\int_0^5 (x(\\sqrt{25-x^2}-(5-x))dx=\\newline\n=\\frac{1}{7}\\int_0^5 (x\\sqrt{25-x^2}-5x+x^2))dx=\\newline\n=\\frac{1}{7}\u2022\\frac{125}{6}=\\frac{125}{42}."

"y_C=\\frac{1}{2S}\\int_0^5(25-x^2-(5-x)^2)dx=\\newline\n\\frac{1}{14}\\int_0^5(-2x^2+10x)dx=\\newline\n\\frac{1}{14}(-\\frac{2}{3}x^3+5x^2)|_0^5=\\newline\n\\frac{1}{14}\u2022\\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})."