# Finding the Length of a Helix Using U-Substitution

• BraedenP
In summary, the conversation discusses a problem involving proving the length of a helix can be represented by a given equation. The attempt at a solution involved using u-substitution, but it was determined that it was not necessary. Instead, the solution involved factoring out a constant and simplifying using trigonometric identities.
BraedenP

## Homework Statement

I am trying to prove that the length of a helix can be represented by $2\pi=\sqrt{a^2+b^2}$

## The Attempt at a Solution

I have the following so far:

If the helix can be represented by $h(t)=a\cdot cos(t)+a\cdot sin(t)+b(t)$

Then the length is:
$$\int_{0}^{2\pi}\sqrt{(-a\cdot sin(t))^2+(a\cdot cos(t))^2+b^2}\;\: dt$$

My problem comes when integrating this. If I use the stuff in the root as u and do u-substitution, then du equals 0dt:

$$u=a^2sin^(t)+a^2cos^2(t)+b^2$$
$$du=(a^2sin(2t)-a^2sin(2t))dt=0dt$$

My logic fails me when figuring out how to continue from there. I need to somehow represent 1dt. How do I do this?

Help would be awesome!

If your du comes out to be zero, then u must be a constant. What constant is it? Use some trig to simplify your u.

You don't need u-substitution for this problem. Here's why:

Before you try taking the integral, inside the square root, you have a2sin2t + a2cos2t. Factor the a2 out of both of them. What happens?

Yep, was definitely over-thinking it. Thanks guys :)

## 1. What is u-substitution and when is it used?

U-substitution is a method used in calculus to simplify integrals by substituting a more complex expression with a simpler one. It is typically used when the integral involves a function within another function, such as f(g(x)).

## 2. How does du=0 impact u-substitution?

When du=0, it means that the derivative of u is equal to 0. This simplifies the integration process as the term involving du can be eliminated from the integral.

## 3. Can any function be used with u-substitution?

No, u-substitution can only be used for integrals where the derivative of u is present in the integrand. Additionally, the substitution must result in a simpler integral to be effective.

## 4. Are there any limitations to using u-substitution?

U-substitution may not work for all integrals, especially those that are more complex. It also may not work for integrals with multiple variables or those that require other integration techniques such as integration by parts.

## 5. How do you know when to use u-substitution in an integral?

If the integral involves a function within another function, it is a good indication that u-substitution may be useful. Additionally, if the integral is in the form of ∫f(g(x))g'(x)dx, then u-substitution can be used.

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