Question

Find the arc length of the curve below on the given interval. x=y^4/4+1/8y^2 for 2 less than or equal to y less than or equal to 3

Answer 1

To find the arc length of the curve
`\displaystyle x=(y^(4))/(4)+(1)/(8)y^(2)` on the interval
`\displaystyle 2\leq y\leq 3`, we can use the arc length formula for a curve given by
`\displaystyle y=f(x)`:

`\displaystyle L=\int _(a)^(b)\sqrt{1+\left( (dy)/(dx)\right)^(2)}\, dx,`

where
`\displaystyle a` and
`\displaystyle b` are the corresponding x-values of the interval
`\displaystyle y=a` and
`\displaystyle y=b`, and
`\displaystyle (dy)/(dx)` is the derivative of
`\displaystyle y` with respect to
`\displaystyle x`.

First, let's find the derivative of
`\displaystyle y` with respect to
`\displaystyle x`:

`\displaystyle (dx)/(dy)=(d)/(dy)\left( (y^(4))/(4)+(1)/(8)y^(2)\right) =y^(3)+(y)/(4)`.

Now, we can calculate the arc length using the given interval
`\displaystyle 2\leq y\leq 3`:

`\displaystyle L=\int _(2)^(3)\sqrt{1+\left( y^(3)+(y)/(4)\right)^(2)}\, dy.`

This integral represents the arc length of the curve. Evaluating this integral will give us the desired result. However, this integral does not have a closed-form solution and must be numerically approximated using methods such as numerical integration or calculus software.