Question

By using the properties of definite integrals, evaluate the integral: \(∫^2_0 x\sqrt{2-x}dx\)

Answer 1

Let I=\(∫^2_0 x\sqrt{2-x}dx\)

\(I=∫^2_0(2-x)\sqrt{x}dx..............(∫^a_0ƒ(x)dx=∫^a_0ƒ(a-x)dx)\)

\(=∫^2_0{2x^\frac{1}{2}-x^\frac{3}{2}}dx\)

\(=[2(\frac{x^\frac{3}{2}}{\frac{3}{2}})\frac{-x^\frac{5}{2}}{\frac{5}{2}}]^2_0\)

\([\frac{4}{3}x^\frac{3}{2}-\frac{2}{5}x^\frac{5}{2}]^2_0\)

\(=\frac{4}{3}(2)^\frac{3}{2}-\frac{2}{5}(2)^\frac{5}{2}\)

\(=\frac{2√2}{3}-\frac{2}{5}×4√2\)

\(=\frac{8√2}{3}-\frac{8√2}{5}\)

\(=\frac{40√2-24√2}{15}\)

\(=\frac{16√2}{15}\)