Question Number 124135 by bramlexs22 last updated on 01/Dec/20
![B = ∫_0 ^2 [(√((4−x)/x)) − (√(x/(4−x))) ] dx](https://www.tinkutara.com/question/Q124135.png)
$$\:{B}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\left[\sqrt{\frac{\mathrm{4}−{x}}{{x}}}\:−\:\sqrt{\frac{{x}}{\mathrm{4}−{x}}}\:\right]\:{dx} \\ $$
Answered by nico last updated on 01/Dec/20
$$=\int_{\mathrm{0}} ^{\mathrm{2}} \frac{\mathrm{4}−{x}−{x}}{\:\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }}{dx}=\int_{\mathrm{0}} ^{\mathrm{2}} \frac{{d}\left(\mathrm{4}{x}−{x}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }} \\ $$$$=\mathrm{2}\sqrt{\mathrm{4}{x}−{x}^{\mathrm{2}} }\mid_{\mathrm{0}} ^{\mathrm{2}} =\mathrm{4} \\ $$