Zadatak 0043

Test: MPsIVKVIV_009

 

Izračunati dužinu dela parabole  \(y=\frac{x^2}{2}\)  na segmentu \(\left [ 0,\sqrt{3} \right ]\).

 

Rešenje:


 

Kako je  \(y'=x\) prema formuli treba izračunati \(L=\int_{0}^{\sqrt{3}}\sqrt{1+x^2}dx\)   

 

\(L=\int_{0}^{\sqrt{3}}\sqrt{1+x^2}dx=\begin{pmatrix}u=\sqrt{1+x^2},du=\frac{x}{\sqrt{1+x^2}}\\v=\int dx=x\end{pmatrix}\)  \(=x\sqrt{1+x^2}\mid _{0}^{\sqrt{3}}-\int_{0}^{\sqrt{3}}\frac{x^2}{\sqrt{1+x^2}}dx\) 

 

\(=2\sqrt{3}-\int_{0}^{\sqrt{3}}\frac{x^2+1-1}{\sqrt{1+x^2}}dx=2\sqrt{3}-\int_{0}^{\sqrt{3}}\sqrt{1+x^2}dx+\int_{0}^{\sqrt{3}}\frac{1}{\sqrt{1+x^2}}dx \) 

 

\(\Rightarrow 2L=2\sqrt{3} +\int_{0}^{\sqrt{3}}\frac{1}{\sqrt{1+x^2}}dx\)\(\Rightarrow L=\sqrt{3}+\frac{1}{2}\ln \left ( x+\sqrt{1+x^2} \right )\mid _{0}^{\sqrt{3}}\)

 

\(=\sqrt{3}+\ln \left ( \sqrt{3}+2 \right )\).