let g(y) = y^4/8 -2+1/4y^2. calculate the arc length of the graph of g(y) over the interval [2,3] enter an exact answer

To find the arc length of the graph of the function \( g(y) \) over the interval \([2, 3]\), we use the arc length formula for a function \( g(y) \) defined with respect to \( y \):

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

Given \( g(y) = \frac{y^4}{8} - 2 + \frac{1}{4}y^2 \), let's first compute the derivative \( \frac{dg}{dy} \).

\[
\frac{dg}{dy} = \frac{d}{dy} \left(\frac{y^4}{8} + \frac{1}{4}y^2 - 2 \right)
\]

Differentiate each term:

\[
\frac{dg}{dy} = \frac{4y^3}{8} + \frac{2y}{4} = \frac{y^3}{2} + \frac{y}{2}
\]

Now, square this derivative:

\[
\left(\frac{dg}{dy}\right)^2 = \left(\frac{y^3}{2} + \frac{y}{2}\right)^2 = \left(\frac{y(y^2 + 1)}{2}\right)^2 = \frac{y^2(y^2 + 1)^2}{4}
\]

Next, substitute into the arc length formula:

\[
L = \int_{2}^{3} \sqrt{1 + \frac{y^2(y^2 + 1)^2}{4}} \, dy
\]

To simplify the integrand:

\[
L = \int_{2}^{3} \sqrt{\frac{4 + y^2(y^2 + 1)^2}{4}} \, dy = \frac{1}{2} \int_{2}^{3} \sqrt{4 + y^2(y^2 + 1)^2} \, dy
\]

This integral is challenging to solve exactly by hand. However, it's clear that the expression under the square root does not simplify nicely, suggesting that the arc length must be expressed as an integral.

So the exact answer is:

\[
L = \frac{1}{2} \int_{2}^{3} \sqrt{4 + y^2(y^2 + 1)^2} \, dy
\]

This is the exact expression for the arc length.

Would you like more details or have any questions?  
Here are 5 related questions:
1. What is the significance of the arc length formula in calculus?
2. How do you solve integrals involving square roots, like the one above?
3. What techniques can simplify complex integrals?
4. How do arc lengths compare for linear vs. nonlinear functions?
5. What are common applications of arc length in real-world problems?

**Tip:** When working with complicated integrals, consider whether numerical methods might be necessary, especially if the integral doesn’t simplify easily.

let g(y) = y^4/8 - 2 + 1/4y^2. calculate the arc length of the graph of g(y) over the interval [2,3] enter an exact answer

Learn how to calculate the arc length of the function g(y) = y^4/8 - 2 + 1/4y^2 over the interval [2,3]. This problem involves calculus concepts, including derivatives, integrals, and the arc length formula. Suitable for college-level calculus students.

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