Question

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

Answer 1

Let I=$∫^2_0 x\sqrt{2-x}dx$

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

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

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

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

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

$=\frac{2√2}{3}-\frac{2}{5}×4√2$

$=\frac{8√2}{3}-\frac{8√2}{5}$

$=\frac{40√2-24√2}{15}$

$=\frac{16√2}{15}$