Find the arc length of the curve below on the given interval.

The length of the curve is
(Type an exact answer, using radicals as needed.)

Ask by Wells Wyatt. in the United States

Mar 07,2025

Question

Find the arc length of the curve below on the given interval. $$ y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}} \text { on }[1,4] $$ The length of the curve is (Type an exact answer, using radicals as needed.)

UpStudy AI · Accepted Answer

To find the arc length of the curve $$ y = \frac{x^4}{4} + \frac{1}{8x^2} $$ on the interval $$[1,4]$$, we can use the formula for the arc length of a curve:

$$ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx $$

First, we need to find the derivative of the function $$ y $$ with respect to $$ x $$:

$$ \frac{dy}{dx} = x^3 - \frac{1}{4x^3} $$

Now, we can substitute this derivative into the formula for the arc length:

$$ L = \int_{1}^{4} \sqrt{1 + \left( x^3 - \frac{1}{4x^3} \right)^2} \, dx $$

Next, we can simplify the expression inside the square root:

$$ L = \int_{1}^{4} \sqrt{1 + x^6 - \frac{1}{2} + \frac{1}{16x^6}} \, dx $$

$$ L = \int_{1}^{4} \sqrt{x^6 + \frac{1}{2} + \frac{1}{16x^6}} \, dx $$

Now, we can calculate the arc length by evaluating the integral.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{1}^{4} \sqrt{x^{6}+\frac{1}{2}+\frac{1}{16x^{6}}} dx$$
- step1: Calculate:
  $$\int_{1}^{4} \left(\left|x^{3}\right|+\frac{1}{4\left|x^{3}\right|}\right) dx$$
- step2: Rewrite the expression:
  $$\int_{1}^{4} \left(x^{3}+\frac{1}{4x^{3}}\right) dx$$
- step3: Evaluate the integral:
  $$\int \left(x^{3}+\frac{1}{4x^{3}}\right) dx$$
- step4: Use properties of integrals:
  $$\int x^{3} dx+\int \frac{1}{4x^{3}} dx$$
- step5: Evaluate the integral:
  $$\frac{x^{4}}{4}+\int \frac{1}{4x^{3}} dx$$
- step6: Evaluate the integral:
  $$\frac{x^{4}}{4}-\frac{1}{8x^{2}}$$
- step7: Return the limits:
  $$\left(\frac{x^{4}}{4}-\frac{1}{8x^{2}}\right)\bigg |_{1}^{4}$$
- step8: Calculate the value:
  $$\frac{8175}{128}$$
The arc length of the curve $$ y = \frac{x^4}{4} + \frac{1}{8x^2} $$ on the interval $$[1,4]$$ is $$ \frac{8175}{128} $$ or $$ 63\frac{111}{128} $$ or approximately $$ 63.8671875 $$.
The arc length of the curve $$ y = \frac{x^4}{4} + \frac{1}{8x^2} $$ on the interval $$[1,4]$$ is $$ \frac{8175}{128} $$ or approximately $$ 63.867 $$.

Find the arc length of the curve below on the given interval.

The length of the curve is
(Type an exact answer, using radicals as needed.)

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Find the arc length of the curve below on the given interval. The length of the curve is (Type an exact answer, using radicals as needed.)