Question

Evaluate the Integral

Answer 1

\[\int\limits_{0}^{10} \sqrt{10x-x^2}dx\]

Answer 2

hint: complete the square

Answer 3

\[10x-x^2=25-25+10x-x^2=25-(5-x)^2\]
\[\int_0^{10}\sqrt{25-(5-x)^2}~dx\]
Then substitute \(5-x=5\sin u\), so that \(dx=-5\cos u~du\):
\[\int_{\pi/2}^{3\pi/2}\sqrt{25-\left(5\sin u\right)^2}~du\]