Find the arc length given the equation y=(x^4/8)+(1/4x^2) [1,3]

To find the arc length of a curve given by an equation, we can use the formula for arc length, which is given by the integral of the square root of the sum of the squares of the derivatives of the function with respect to x, multiplied by dx.

Let's break down the steps to find the arc length of the curve defined by the equation y = (x^4/8) + (1/4x^2) over the interval [1,3].

Step 1: Find the derivative of the function y(x) with respect to x.

Taking the derivative of y(x) = (x^4/8) + (1/4x^2) gives us:
dy/dx = (4x^3)/8 - (1/2x^3)

Simplifying, we get:
dy/dx = x^3/2 - 1/2x^3

Step 2: Square the derivative and take the square root.

Taking the square of the derivative, we have:
(dy/dx)^2 = (x^3/2 - 1/2x^3)^2
= [(x^6)/4 - x^3 + (1/4)(1/x^6)]

Taking the square root gives:
sqrt((dy/dx)^2) = sqrt[(x^6)/4 - x^3 + (1/4)(1/x^6)]

Step 3: Set up the integral.

The arc length integral is given by:
L = ∫[a,b]sqrt(1 + (dy/dx)^2)dx

Substituting the expression for sqrt((dy/dx)^2), we get:
L = ∫[a,b]sqrt[1 + (x^6)/4 - x^3 + (1/4)(1/x^6)]dx

Step 4: Evaluate the integral.

Now we can evaluate the integral from the lower limit a=1 to the upper limit b=3. However, it's worth noting that finding the exact solution to this integral is challenging and usually requires numerical methods or specialized software.

If you have access to calculus software or a graphing calculator, you can input the integrand and evaluate the integral directly.

Alternatively, you can approximate the arc length by using numerical methods such as the trapezoidal rule or Simpson's rule.

In summary, to find the arc length of the curve defined by the equation y = (x^4/8) + (1/4x^2) over the interval [1,3], we need to evaluate the integral of sqrt(1 + (dy/dx)^2)dx from 1 to 3. This can be done using calculus software or numerical methods.