## Introduction

In this article, we will be looking at how to find the exact length of the curve x(t)=e^(4t) and y(t)=8e^(2t) for 0<=t<=3. We will first look at finding the length of the curve using a numerical method, and then we will look at how to find the exact length of the curve using Calculus.

## What is the Length of a Curve?

Many different formulas can be used to calculate the length of a curve. In general, the length of a curve can be found by integrating the curve’s velocity over time. However, the specific method that you use will depend on the type of curve that you are dealing with.

## How to Find the Length of a Curve

There are a few different ways to find the length of a curve, but the most common is to use the arc length formula. This formula allows you to find the length of any curve by summing up the lengths of small segments of the curve. To use this formula, you need to know the coordinates of points on the curve and the value of t (the parameter) at those points.

## Examples

Here are some examples of finding the exact length of a curve:

x et 4t y 8et2 0 t 3

First, we’ll take the derivative of both x and y with respect to t:

dx/dt = e^t(4t + 4)

dy/dt = 8e^(2t)(2t + 1)

Then, we’ll take the square root of the sum of the squares of dx/dt and dy/dt:

sqrt((dx/dt)^2 + (dy/dt)^2) = sqrt((e^t(4t + 4))^2 + (8e^(2t)(2t + 1))^2)

Finally, we’ll integrate both sides with respect to t from 0 to 3:

int_0^3 sqrt((dx/dt)^2 + (dy/dt)^2) dt = int_0^3 sqrt((e^t(4t + 4))^2 + (8e^(2t)(2t + 1))^2) dt

## Conclusion

In conclusion, to find the exact length of the curve x et 4t y 8et2 0 t 3, we must first calculate the integrand. The integrand is a function of t and is defined as x et 4t y 8et2 0 t 3. The integrand will be used to calculate the exact length of the curve. The exact length of the curve can be found by using the following definite integral:

definite integral of (x*sqrt(1+(dy/dx)^2)) from a to b

The above definite integral will give us the exact length of the curve if we can calculate the value of the integrand at various points along the curve. In order to do this, we must first find the derivative of the integrand with respect to t. The derivative of the integrand with respect to t is given by:

derivative of (x*sqrt(1+(dy/dx)^2)) with respect to t = (sqrt(1+(dy/dx)^2))+(x*((dy/dx)*(1+(dy/dx)^2)^(-1/2)))*(derivative of (dy/dx) with respect to t)

Since we are only interested in finding the value of the derivative at specific points along the curve, we can plug in values for x and y in terms of t. Plugging in values for x and y in terms of t, we get:

derivative of (x*sqrt(1+(dy/dx)^2)) with respect to t = (sqrt(1+((4t-8t^3)/(4t^2-12t+9))^2))+((4t-8t^3)/(4t^3-36t+27))*((4-24t+36t)/(4t^3-36))