# Finding Arc Length of x = (y^4)/8 + 1/(4y^2) from 1 to 2

• MHrtz
In summary, to determine the arc length of the given function on the interval 1 to 2, you can use the arc length formula \int (f'(x)2 + 1).5 dx. In this case, it is helpful to use negative exponents instead of fractions. Factoring and taking the square root of the resulting perfect square, you will get (y^3)/2 + 1/2y^3. This can then be integrated to find the arc length.
MHrtz

## Homework Statement

Determine the arc length of the function on the given interval

x = (y^4)/8 + 1/(4y^2) from 1 to 2

The arc length formula

$\int$ (f'(x)2 + 1).5 dx

## The Attempt at a Solution

I used the arc length formula but don't know where to go from here. Usually these problems can't be done unless a form of cancellation takes place and I can't seem to find it. Below is what I input in the formula but I can not figure out how to integrate this.

$\int$((y6 - 1)2/(2y3)) + 1))).5 dy

MHrtz said:

## Homework Statement

Determine the arc length of the function on the given interval

x = (y^4)/8 + 1/(4y^2) from 1 to 2

The arc length formula

$\int$ (f'(x)2 + 1).5 dx

## The Attempt at a Solution

I used the arc length formula but don't know where to go from here. Usually these problems can't be done unless a form of cancellation takes place and I can't seem to find it. Below is what I input in the formula but I can not figure out how to integrate this.

$\int$((y6 - 1)2/(2y3)) + 1))).5 dy

It's helpful in this problem to use negative exponents instead of fractions.

(x')2 + 1 = (1/4)y6 - 1/2 + (1/4)y-6 + 1
= (1/4)y6 + 1/2 + (1/4)y-6

This turns out to be a perfect square, so you can readily take its square root.

I factored and then took the square root and came up with this:

(y^3)/2 + 1/2y^3

Yes, that is correct. Now integrate.

## 1. What is the formula for finding arc length?

The formula for finding arc length is L = ∫√(1 + (dy/dx)^2)dx, where dx is the independent variable and dy/dx is the derivative of the function.

## 2. How do you find the derivative of the given function?

To find the derivative of x = (y^4)/8 + 1/(4y^2), we use the power rule and the chain rule. The derivative is given by dy/dx = (4y^3)/8 - (1/2)(4y^-3)(dy/dx).

## 3. What are the limits of integration for this function?

The limits of integration for this function are 1 and 2, as given in the problem.

## 4. Can this formula be used for any type of function?

Yes, the formula for finding arc length can be used for any type of function, as long as it is continuous and differentiable within the given limits of integration.

## 5. Is there a simpler way to find arc length?

Yes, there is a simpler way to find arc length for certain functions using the arc length formula for a circle, L = 2πrθ, where r is the radius and θ is the angle subtended by the arc. However, this can only be used for functions that can be written in the form y = f(x) or x = g(y).

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