## Introduction

In order to find the length of a curve, we need to first have its equation in the form of either y=f(x) or x=g(y). In this case, we have x=et and y=4t, so we’ll use the latter form. We’ll also need the limits of integration, which in this case are 0 and 2.

Now that we have all the necessary information, we can begin finding the length of the curve using the following formula:

L = ∫sqrt(1 + (dy/dx)^2)dx

Applying this to our specific case, we get:

L = ∫sqrt(1 + (4/e^t)^2)det

Which can be simplified to:

L = ∫sqrt((e^t + 4/e^t)^2)det

After some further simplification and algebraic manipulation, we arrive at:

L = 2∫e^t(1 + 1/4e^{2t})dt 0<= t<= 2

## The Length of a Curve

Given a curve defined by the equation x = et – 4t, y = 8et2 – 0t, 0 < t < 2, find the exact length of the curve.

First, we need to find the derivatives of x and y with respect to t.

x’(t) = ett – 4

y’(t) = 16e2t – 0

Next, we need to find the derivative of √(x’(t))2 + (y’(t))2 with respect to t.

√(x’(t))2 + (y’(t))2 = √((ett – 4)2 + (16e2t – 0)2)

= √((ett)2 + (-4)2 + (16e2t)2 + (0)2)

= √(e4t + 16e4t + 16).

= √(17e4t).

= e2t√17.

Now we can integrate both sides of this equation from 0 to 2 to find the length of the curve.

∫0∫2Length of Curve ds dT = ∫0∫217e4tdtdT

= [17e4T]022

= 340e8.

## The Exact Length of the Curve

If you’re looking to find the exact length of the curve x et 4t y 8et2 0 t 2, there are a few things you’ll need to know. First, you’ll need to know the definition of the curve. A curve is a line that is not straight – it bends and curves. In order to find the length of this curve, we will need to use a formula. The formula for finding the length of a curve is:

length = integral of (1 + (dy/dx)^2)^(1/2) from a to b

In order to use this formula, we will need to know the value of dy/dx. We can find this by taking the derivative of both sides with respect to x. This gives us:

dy/dx = (4t)/(8et^2)

Now that we have dy/dx, we can plug it into our formula for length. We will also need to know the value of t when x=0 and when x=2. We can plug these values in and solve for the length of the curve:

length = integral of (1 + ((4t)/(8et^2))^2)^(1/2) from 0 to 2

After solving this integral, we get a length of approximately 4.7 units.

## Conclusion

To find the exact length of the curve, we need to find the derivative of x and y with respect to t.

Differentiating x with respect to t, we get: dx/dt = e^t(4t)

Differentiating y with respect to t, we get: dy/dt = 8e^(2t)(2t)

Therefore, the exact length of the curve is given by: L = int_0^2sqrt((dx/dt)^2 + (dy/dt)^2) dt

= int_0^2sqrt(e^(2t)(4t)^2 + 8e^(4t)(2t)^2) dt

= int_0^2sqrt(16e^(2t)(t^2 + 2t + 1)) dt

= int_0^2sqrt(16e^(2r))*((r+1/2)+(1/8)) dr = 2sqrt(16)*int_0^1sqrt((r+1/2)+1/8)) dr (let u= r+1/20, then du= dr )

```
= 2sqrt{16}*int_9/20*sqrt{u+(1-9/(20*u)} du
= 2sqrt{16}*[u*sqart{u+(1-9/(20*u)}]_9/20 1
= 2 * 4 * sqart{80} * [u* sqart { u+4/(5u)}] _ 9 / 20 1
= 25.6 * [ u * sqart { u+ 4 / (5u)}] _ 9 / 20 1</p>
```