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## Homework Statement

Find area bounded by functions [itex]y_1=\sqrt{4x-x^2}[/itex] and [itex]y_2=x\sqrt{4x-x^2}[/itex].

## Homework Equations

-Integration

-Area

## The Attempt at a Solution

From [itex]y_1=y_2\Rightarrow x=1[/itex]. Intersection points of [itex]y_1[/itex] and [/itex]y_2[/itex] are [itex]A(0,0),B(1,\sqrt 3),C(4,0)[/itex]. Domain of [itex]y_1[/itex] and [itex]y_2[/itex] is [itex]x\in [0,4][/itex]. On the interval [itex]x\in[0,1]\Rightarrow y_1\ge y_2[/itex] and on the interval [itex]x\in[1,4]\Rightarrow y_1\le y_2[/itex].

[itex]A=\int_0^1 (y_1-y_2)\mathrm dx+\int_1^4 (y_2-y_1)\mathrm dx=\int_0^1 (1-x)\sqrt{4x-x^2}\mathrm dx+\int_1^4 (x-1)\sqrt{4x-x^2}\mathrm dx[/itex]

How to solve integrals [itex]\int \sqrt{4x-x^2}\mathrm dx[/itex] and [itex]\int x\sqrt{4x-x^2}\mathrm dx[/itex]?

Substitution [itex]u=\sqrt{\frac{x}{4-x}}\Rightarrow du=\frac{2}{(x-4)^2\sqrt{\frac{x}{4-x}}}dx[/itex] doesn't seems to work.