Question

Determine the length of the curve $y=\frac{x^2}{2}+\frac{\log x}{4}$ within $2\le x\le 4$.

• A. $8+\frac{\log 2}{4}$
• B. $6+\frac{\log 2}{4}$
• C. $6+\frac{\log 2}{3}$
• D. $3+\frac{\log 2}{4}$

Answer 1

The correct option is B $6+\frac{\log 2}{4}$

For a function f(x), length of the curve in the interval $(a,b)$ is given by

${\int }_a^b{f}^{{}^{\prime }}\left(x\right)dx$

Here, $f\left(x\right)=y=\frac{x^2}{2}+\frac{\log x}{4}$

$⟹f^{{}^{\prime }}\left(x\right)=2x+\frac{1}{4x}$

Now, length of the curve in the interval $[2,4]$ is given by

${\int }_2^4{f}^{{}^{\prime }}\left(x\right)dx={\int }_2^4(2x+\frac{1}{4x})dx$

$={[\frac{x^2}{2}+\frac{1}{4}\log x]}_2^4$

$=6+\frac{\log 2}{4}$

Hence , answer is option-(B).