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) $\frac { \ln 2 } { 2 }$
B) $3 + \ln 2$
C) $2 + \ln 3$
D) $3 + \frac { \ln 2 } { 2 }$

Question

Quizplus · Accepted Answer

The length $L$ of a curve $y = f(x)$ from $x = a$ to $x = b$ is given by the formula $L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$. For $y = \frac{x^2}{4} - \frac{1}{2}\ln x$, we first find the derivative $y'$: $y' = \frac{x}{2} - \frac{1}{2x}$. Squaring this and adding 1 gives $1 + \left(\frac{x}{2} - \frac{1}{2x}\right)^2 = 1 + \frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2} = \frac{x^2}{4} + \frac{1}{2} + \frac{1}{4x^2}$. Integrating this expression from 2 to 4, we get the length of the curve. The integral simplifies to integrating $\frac{x^2 + 2x^{-2}}{4} + \frac{1}{2}$, which results in $\frac{x^3}{12} - \frac{1}{2x} + \frac{x}{2}$ evaluated from 2 to 4. This calculation yields $3 + \frac{\ln 2}{2}$, which corresponds to choice D.

Solve the Problem y=x24−12ln⁡x,2≤x≤4y = \frac { x ^ { 2 } } { 4 } - \frac { 1 } { 2 } \ln x , 2 \leq x \leq 4y=4x2​−21​lnx,2≤x≤4

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