Integrals — Maths STD 12 Science — Question

$= \int\limits_0^2 {\left( {2 - x} \right)\sqrt {2 - \left( {2 - x} \right)} dx} $

$\left[ {\because \int\limits_0^a {f\left( x \right)dx = \int\limits_0^a {f\left( {a - x} \right)dx } } } \right]$

$\Rightarrow I = \int\limits_0^2 {\left( {2 - x} \right)\sqrt x dx} $

$= \int\limits_0^2 {\left( {2{x^{\frac{1}{2}}} - {x^{\frac{3}{2}}}} \right)dx} $

$= \left[ {2.\frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}} - \frac{{{x^{\frac{5}{2}}}}}{{\frac{5}{2}}}} \right]_0^2$

$= \left( {\frac{4}{3}{{.2}^{\frac{3}{2}}} - \frac{2}{5}{{.2}^{\frac{5}{2}}}} \right) - \left( {0 - 0} \right)$

$\Rightarrow I = \frac{4}{3} \times 2\sqrt 2 - \frac{2}{5} \times 4\sqrt 2 $

$ = \left( {\frac{8}{3} - \frac{8}{5}} \right)\sqrt 2 $

$= \frac{{16\sqrt 2 }}{{15}}$