Find the length of the curve.
-$$y = \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 2 x } \text { from } x = 1 \text { to } x = 5$$
A) $\frac { 316 } { 15 }$
B) $\frac { 127 } { 6 }$
C) $\frac { 79 } { 5 }$
D) $\frac { 632 } { 15 }$

Question

Quizplus · Accepted Answer

The length $L$ of a curve defined by $y = f(x)$ from $x = a$ to $x = b$ can be found using the formula $$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx.$$ For the given function $y = \frac{1}{6}x^3 + \frac{1}{2x},$ first, we find its derivative with respect to $x$: $$\frac{dy}{dx} = \frac{1}{2}x^2 - \frac{1}{2x^2}.$$ Squaring this derivative and adding 1 gives: $$1 + \left(\frac{1}{2}x^2 - \frac{1}{2x^2}\right)^2 = 1 + \frac{1}{4}x^4 - \frac{1}{2} + \frac{1}{4x^4} = \frac{1}{4}x^4 + \frac{1}{2} + \frac{1}{4x^4}.$$ Integrating this expression from $x = 1$ to $x = 5$, we get: $$L = \int_{1}^{5} \sqrt{\frac{1}{4}x^4 + \frac{1}{2} + \frac{1}{4x^4}} dx.$$ Simplifying and solving this integral, we find that the length of the curve is $\frac{316}{15},$ which corresponds to choice A.

Find the Length of the Curve y=16x3+12x from x=1 to x=5y = \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 2 x } \text { from } x = 1 \text { to } x = 5y=61​x3+2x1​ from x=1 to x=5

Find the Length of the Curve $y = \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 2 x } \text { from } x = 1 \text { to } x = 5$