Mathematics

Solve $$\int\limits_2^6 {\sqrt {\left( {6 - x} \right)\left( {x - 2} \right)} dx} $$


SOLUTION

We have,

$$I=\int_{2}^{6}{\sqrt{\left( 6-x \right)\left( x-2 \right)}}\,dx$$

$$ I=\int_{2}^{6}{\sqrt{6x-12-{{x}^{2}}+2x}}\,dx $$

$$ I=\int_{2}^{6}{\sqrt{8x-12-{{x}^{2}}}}\,dx $$

$$ I=\int_{2}^{6}{\sqrt{8x-12-16+16-{{x}^{2}}}}\,dx $$

$$ I=\int_{2}^{6}{\sqrt{{{\left( 2\sqrt{7} \right)}^{2}}-{{\left( 4-x \right)}^{2}}}}\,dx $$

 

Let

$$t=4-x$$

$$-dt=dx$$

 

Therefore,

$$I=-\int_{2}^{-2}{\sqrt{{{\left( 2\sqrt{7} \right)}^{2}}-{{t}^{2}}}}\,dt$$

 

We know that

$$\int{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\dfrac{1}{2}x\sqrt{{{a}^{2}}-{{x}^{2}}}+\dfrac{1}{2}{{a}^{2}}{{\sin }^{-1}}\left( \dfrac{x}{a} \right)+C$$

 

Therefore,

$$ I=-\left[ \dfrac{1}{2}t\sqrt{{{\left( 2\sqrt{7} \right)}^{2}}-{{t}^{2}}}+\dfrac{1}{2}{{\left( 2\sqrt{7} \right)}^{2}}{{\sin }^{-1}}\left( \dfrac{t}{2\sqrt{7}} \right) \right]_{2}^{-2} $$

$$=-\left( \dfrac{1}{2}\left( -2 \right)\sqrt{{{\left( 2\sqrt{7} \right)}^{2}}-{{\left( -2 \right)}^{2}}}+\dfrac{1}{2}{{\left( 2\sqrt{7} \right)}^{2}}{{\sin }^{-1}}\left( \dfrac{-2}{2\sqrt{7}} \right) \right)+$$

         $$\left( \dfrac{1}{2}\left( 2 \right)\sqrt{{{\left( 2\sqrt{7} \right)}^{2}}-{{\left( 2 \right)}^{2}}}+\dfrac{1}{2}{{\left( 2\sqrt{7} \right)}^{2}}{{\sin }^{-1}}\left( \dfrac{2}{2\sqrt{7}} \right) \right)$$


$$ I=- \left( -\sqrt{28-4}+\dfrac{1}{2}\times 28{{\sin }^{-1}}\left( \dfrac{-1}{\sqrt{7}} \right) \right)+\left( \sqrt{28-4}+\dfrac{1}{2}\times 28{{\sin }^{-1}}\left( \dfrac{1}{\sqrt{7}} \right) \right)$$

$$ I=-\left( -\sqrt{24}+14{{\sin }^{-1}}\left( \dfrac{-1}{\sqrt{7}} \right) \right)+\left( \sqrt{24}+14{{\sin }^{-1}}\left( \dfrac{1}{\sqrt{7}} \right) \right)$$

$$ I=2\sqrt{24}-14{{\sin }^{-1}}\left( \dfrac{-1}{\sqrt{7}} \right)+14{{\sin }^{-1}}\left( \dfrac{1}{\sqrt{7}} \right) $$

 

Hence, the value is $$2\sqrt{24}-14{{\sin }^{-1}}\left( \dfrac{-1}{\sqrt{7}} \right)+14{{\sin }^{-1}}\left( \dfrac{1}{\sqrt{7}} \right)$$.

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