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Question Number 204802 by Faetmaaa last updated on 27/Feb/24
Wi-Fi code problem: ∫_(−2) ^( 2) (x^3 cos((x/2))+(1/2))(√(4−x^2 ))dx
$$\mathrm{Wi}-\mathrm{Fi}\:\mathrm{code}\:\mathrm{problem}: \\ $$$$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \left({x}^{\mathrm{3}} \mathrm{cos}\left(\frac{{x}}{\mathrm{2}}\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\mathrm{d}{x} \\ $$
Answered by TonyCWX08 last updated on 28/Feb/24
Expand The Integral ∫_(−2) ^2 (x^3 cos ((x/2))(√(4−x^2 )))dx+(1/2)∫_(−2) ^2 (√(4−x^2 ))dx Let x^3 cos ((x/2))(√(4−x^2 )) = f(x) Observe That f(−x) = (−x)^3 cos (((−x)/2))(√(4−(−x)^2 )) = −x^3 cos((x/2))(√(4−x^2 )) =−f(x) Conclusion: f(x) Is An Odd Function Know ∫_(−a) ^a (Odd Function)dx = 0 ∫_(−2) ^2 (x^3 cos ((x/2))(√(4−x^2 )))dx = 0 0+(1/2)∫_(−2) ^2 (√(4−x^2 ))dx Observe That (√(4−x^2 ))Is A Semicircle With Radius 2 So, Area Of The Semicircle = (1/2)π(2)^2 =2π 0+(1/2)(2π)=π
$$ \\ $$$${Expand}\:{The}\:{Integral} \\ $$$$\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\left({x}^{\mathrm{3}} \mathrm{cos}\:\left(\frac{{x}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\right){dx}+\frac{\mathrm{1}}{\mathrm{2}}\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }{dx} \\ $$$$ \\ $$$${Let}\:{x}^{\mathrm{3}} \mathrm{cos}\:\left(\frac{{x}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:=\:{f}\left({x}\right) \\ $$$${Observe}\:{That} \\ $$$${f}\left(−{x}\right)\:=\:\left(−{x}\right)^{\mathrm{3}} \mathrm{cos}\:\left(\frac{−{x}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−\left(−{x}\right)^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:−{x}^{\mathrm{3}} {cos}\left(\frac{{x}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=−{f}\left({x}\right) \\ $$$${Conclusion}:\:{f}\left({x}\right)\:{Is}\:{An}\:{Odd}\:{Function} \\ $$$${Know}\:\underset{−{a}} {\overset{{a}} {\int}}\left({Odd}\:{Function}\right){dx}\:=\:\mathrm{0} \\ $$$$\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\left({x}^{\mathrm{3}} \mathrm{cos}\:\left(\frac{{x}}{\mathrm{2}}\right)\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\right){dx}\:=\:\mathrm{0} \\ $$$$ \\ $$$$\mathrm{0}+\frac{\mathrm{1}}{\mathrm{2}}\underset{−\mathrm{2}} {\overset{\mathrm{2}} {\int}}\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }{dx} \\ $$$${Observe}\:{That} \\ $$$$\sqrt{\mathrm{4}−{x}^{\mathrm{2}} \:\:}{Is}\:{A}\:{Semicircle}\:{With}\:{Radius}\:\mathrm{2} \\ $$$${So},\:{Area}\:{Of}\:{The}\:{Semicircle}\:=\:\frac{\mathrm{1}}{\mathrm{2}}\pi\left(\mathrm{2}\right)^{\mathrm{2}} =\mathrm{2}\pi \\ $$$$\mathrm{0}+\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{2}\pi\right)=\pi \\ $$

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