The arc length of $$ f(x)=\frac{1}{8} x^{2}-\ln x $$ from $$ x=1 $$ to $$ x=2 $$ is $$ \begin{array}{lll} ext { A. } \frac{1}{3} \ln 2 & ext { B. } \frac{1}{8} \ln 3 & ext { C. } \frac{1}{4}+\frac{1}{2} \ln 2 \ \frac{3}{8}+\ln 2 & \frac{1}{8}+\ln 2\end{array} $$

Question

The arc length of $$ f(x)=\frac{1}{8} x^{2}-\ln x $$ from $$ x=1 $$ to $$ x=2 $$ is $$ \begin{array}{lll}	ext { A. } \frac{1}{3} \ln 2 & 	ext { B. } \frac{1}{8} \ln 3 & 	ext { C. } \frac{1}{4}+\frac{1}{2} \ln 2 \ \frac{3}{8}+\ln 2 & \frac{1}{8}+\ln 2\end{array} $$

UpStudy AI · Accepted Answer

The calculation of the integral seems to be too complex and has timed out. Let's simplify the integral before calculating it.

The integral is:

$$ L = \int_{1}^{2} \sqrt{1 + \left( \frac{1}{4}x - \frac{1}{x} \right)^2} dx $$

We can simplify the expression inside the square root:

$$ 1 + \left( \frac{1}{4}x - \frac{1}{x} \right)^2 = 1 + \frac{1}{16}x^2 - \frac{1}{2} + \frac{1}{x^2} $$

$$ = \frac{1}{16}x^2 + \frac{1}{2} + \frac{1}{x^2} $$

Now, we can substitute this simplified expression back into the integral:

$$ L = \int_{1}^{2} \sqrt{\frac{1}{16}x^2 + \frac{1}{2} + \frac{1}{x^2}} dx $$

Let's calculate the integral again.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{1}^{2} \sqrt{\frac{1}{16}x^{2}+\frac{1}{2}+\frac{1}{x^{2}}} dx$$
- step1: Calculate:
  $$\int_{1}^{2} \left(\frac{\left|x\right|}{4}+\frac{1}{\left|x\right|}\right) dx$$
- step2: Rewrite the expression:
  $$\int_{1}^{2} \left(\frac{x}{4}+\frac{1}{x}\right) dx$$
- step3: Evaluate the integral:
  $$\int \left(\frac{x}{4}+\frac{1}{x}\right) dx$$
- step4: Use properties of integrals:
  $$\int \frac{x}{4} dx+\int \frac{1}{x} dx$$
- step5: Evaluate the integral:
  $$\frac{x^{2}}{8}+\int \frac{1}{x} dx$$
- step6: Evaluate the integral:
  $$\frac{x^{2}}{8}+\ln{\left(\left|x\right|\right)}$$
- step7: Return the limits:
  $$\left(\frac{x^{2}}{8}+\ln{\left(\left|x\right|\right)}\right)\bigg |_{1}^{2}$$
- step8: Calculate the value:
  $$\frac{3+8\ln{\left(2\right)}}{8}$$
The arc length of the function $$ f(x) = \frac{1}{8}x^{2} - \ln x $$ from $$ x = 1 $$ to $$ x = 2 $$ is $$ \frac{3+8\ln{(2)}}{8} $$.

Therefore, the correct answer is $$ \frac{3+8\ln{(2)}}{8} $$.
The arc length is $$ \frac{3}{8} + \ln 2 $$.

The arc length of from to is

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