# Solving Arclength of Curve r(t) | Step-by-Step Guide

• olivia333
In summary: W1pbmFyeSwsIFwiSW4gc3VtbWFyeSwgXCJBbmQgb3V0bG9vaW5nIHlvdSBpbiB0aGUgbGFzdCBzdGVwLCBidXQgSSd2ZSBjaG9vc2Ugb24gdGhlIGxhc3Qgc3RlcCwgYnV0IGlubmVyZ3kgb2YgdGhlIGxhc3Qgc3RlcC4gVGhlcmUgaXQncyBhIGNvbXBsZWQgc3RlcHMuIFRoZW4gdGhlIHVuc2Fm
olivia333

## Homework Statement

I have the problem and answer, I'm just confused on the last step, but I'll put down everything anyways.

Find the arclength of the curve r(t) = <5√(2)t, e5t, e-5t>
0≤t≤1

L = ∫|r'(t)|

## The Attempt at a Solution

r'(t) = √((5√(2))2+(5e5t)2+(-5e-5t)2)

|r'(t)| = 5√(2+e10t+e-10t)

Now this is where I get confused.

∫(5√(2+e10t+e-10t)) = ((e10t-1)*√(2+e10t+e-10t))/(e10t+1)

I just don't understand how to integrate my answer to get that. If someone could just go through the steps that'd be great. Thanks!

Then obviously sub in for 1 and 0, getting 148.4064 which is correct so I know that the integration is correctly done.

Last edited:
That's an odd form for the resulting integral. To see how to do it more easily what is (e^(5t)+e^(-5t))^2? Expand it out.

olivia333 said:

## Homework Statement

I have the problem and answer, I'm just confused on the last step, but I'll put down everything anyways.

Find the arclength of the curve r(t) = <5√(2)t, e5t, e-5t>
0≤t≤1

L = ∫|r'(t)|

## The Attempt at a Solution

r'(t) = √((5√(2))2+(5e5t)2+(-5e-5t)2)

|r'(t)| = 5√(2+e10t+e-10t)

Now this is where I get confused.

∫(5√(2+e10t+e-10t)) = ((e10t-1)*√(2+e10t+e-10t))/(e10t+1)

I just don't understand how to integrate my answer to get that. If someone could just go through the steps that'd be great. Thanks!

Then obviously sub in for 1 and 0, getting 148.4064 which is correct so I know that the integration is correctly done.

Your function $5\sqrt{2 + e^{10t} + e^{-10t}}$ can be re-written, using the identity $\cosh(10t) =2 \cosh(5t)^2 - 1,$ to give a much simpler integral.

RGV

Last edited:

## 1. What is the formula for finding the arclength of a curve?

The formula for finding the arclength of a curve is ∫√[1 + (dy/dx)^2]dx, where dy/dx represents the first derivative of the curve.

## 2. How do you find the derivative of a curve?

To find the derivative of a curve, you use the power rule or chain rule, depending on the complexity of the function. The derivative represents the slope of the curve at a specific point.

## 3. What is the significance of finding the arclength of a curve?

Finding the arclength of a curve is important in various fields such as physics, engineering, and mathematics. It allows us to measure the length of a curve, which can help in understanding the shape and behavior of the curve.

## 4. What are the steps for solving the arclength of a curve?

The steps for solving the arclength of a curve are as follows:
1. Find the derivative of the curve
2. Square the derivative and add 1
3. Integrate the resulting expression with respect to x
4. Substitute the limits of integration and evaluate the integral
5. Take the square root of the result to get the arclength of the curve.

## 5. Can the arclength of a curve be negative?

No, the arclength of a curve cannot be negative. The length of a curve is always a positive value, and the arclength formula takes this into account by taking the absolute value of the derivative.

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