Question

$\pi<\frac{22}{7}$ olduğunu gösteriniz.

Answer 1

$$x\in (0,1)$$ $$\Rightarrow$$ $$0<\frac{x^4(1-x)^4}{1+x^2}$$ $$\Rightarrow$$ $$0<\int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} dx$$ $$=$$ $$\int_{0}^{1} \frac{x^4 (1-4x+6x^2-4x^3+x^4)}{1+x^2} dx$$ $$=$$ $$\int_{0}^{1} \frac{x^4 -4x^5+6x^6-4x^7+x^8}{1+x^2} dx$$ $$=$$ $$\int_{0}^{1}\left(x^6-4x^5+5x^4-4x^2+4-\frac{4}{1+x^2}\right)dx$$ $$=$$ $$\left(\frac{x^7}{7}-\frac{2x^6}{3}+x^5-\frac{4x^3}{3}+4x-4\arctan x\right)\Big{|}_{0}^{1}$$ $$=$$ $$\frac{22}{7}-\pi$$ $$\Rightarrow$$ $$\pi<\frac{22}{7}$$

Answer 2

$\pi <3,142<\frac{22}{7}$.

Comments

:-) :-) :-) :-)

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bomba gibi bir cevap hocam  yalın ve net :-)))