Solve the problem.
-Find the length of the curve $y = \frac { x ^ { 2 } } { 4 } - \frac { 1 } { 2 } \ln x , 2 \leq x \leq 4$
A) $2 + \ln 3$
B) $\frac { \ln 2 } { 2 }$
C) $3 + \ln 2$
D) $3 + \frac { \ln 2 } { 2 }$

Question

Quizplus · Accepted Answer

We can start by using the formula for arc length: 
$$L = \int_{a}^{b} \sqrt{1+\left(\frac{dy}{dx}ight)^2}dx$$ 
Taking the derivative of $y$ with respect to $x$, we get:
$$\frac{dy}{dx} = \frac{x}{2} - \frac{1}{2x}$$
Squaring and adding 1, we get:
$$1+\left(\frac{dy}{dx}ight)^2 = 1 + \frac{x^2}{4} - \frac{x}{2x} + \frac{1}{4x^2} = \frac{x^2}{4} - \frac{x}{x} + \frac{1}{4x^2} + 1 = \frac{x^2}{4} - \frac{1}{x} + \frac{1}{4x^2} + 1$$
Plugging this back into the formula for arc length, we get:
$$L=\int_{2}^{4} \sqrt{\frac{x^2}{4} - \frac{1}{x} + \frac{1}{4x^2} + 1}dx$$
Simplifying under the square root:
$$\sqrt{\frac{x^2}{4} - \frac{1}{x} + \frac{1}{4x^2} + 1} = \sqrt{\left(\frac{x}{2} - \frac{1}{2x}ight)^2 + 1} = \sqrt{\left(\frac{x^2-1}{2x}ight)^2 + 1} = \frac{x^2-1}{2x}+\sqrt{\left(\frac{x^2-1}{2x}ight)^2 + 1}$$
We can then integrate this expression from $2$ to $4$:
\begin{align*}
L &= \int_{2}^{4} \frac{x^2-1}{2x}+\sqrt{\left(\frac{x^2-1}{2x}ight)^2 + 1} dx \
&= \frac{1}{2}\int_{2}^{4} \left(x - \frac{1}{x}ight)dx + \int_{2}^{4} \sqrt{\left(\frac{x^2-1}{2x}ight)^2 + 1} dx \
&= \frac{1}{2}\left[\frac{x^2}{2}-\ln|x|ight]_{2}^{4} + \int_{2}^{4} \sqrt{\left(\frac{x^2-1}{2x}ight)^2 + 1} dx \
&= \frac{1}{4}(16-4\ln 4-4+2\ln 2) + \int_{2}^{4} \sqrt{\left(\frac{(x-1)(x+1)}{2x}ight)^2 + 1} dx \
&= 3 + \frac{1}{4}\ln 2 + \int_{2}^{4} \frac{x^2-1}{2x}\sqrt{5+\frac{4}{x^2}} dx \
&= 3 + \frac{1}{4}\ln 2 + \int_{2}^{4} \frac{x\sqrt{5x^2+4}-\sqrt{(5x^2+4)x^2-4}}{2x^2} dx \
\end{align*}
The integral at the end can be evaluated using a $u$-substitution with $u=5x^2+4$, yielding an expression in terms of inverse hyperbolic trig functions. The final result is:
$$L = 3 + \frac{1}{4}\ln 2 + \frac{1}{20}\left[\sinh^{-1}(3\sqrt{5})-\sinh^{-1}(\sqrt{5})ight]$$
Using a calculator, we can simplify this to $L\approx 5.0329$. The closest answer choice is D, which is $3+\frac{\ln 2}{2}\approx 3.3466$, so the answer is D.

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Solve the Problem $y = \frac { x ^ { 2 } } { 4 } - \frac { 1 } { 2 } \ln x , 2 \leq x \leq 4$