# Length of Curve: Evaluating the Integral

• lmh4
In summary, the length of a curve is the distance between two points on the curve and is measured using integration. The formula for finding the length of a curve is L = ∫√(1+ (dy/dx)^2) dx. It has various applications in different fields, but there are limitations to using integration for finding the length of a curve, such as only being applicable to continuous curves. Other methods may be used for curves with sharp turns or corners.

## Homework Statement

Find the length of the curve.

x=cos(2t)
y=sin(3t)
0≤t≤2∏

I know the length is just the integral from 0 to 2∏of the magnitude of the velocity.

## The Attempt at a Solution

x'=-2sin(2t) x'^2=4sin^2(2t)
y'=3cos(3t) y'^2=9cos^2(3t)

c=∫√4sin^2(2t)+9cos^2(3t)

Im having trouble evaluating the integral. Could some please point me in the right direction?

Thanks

$4sin^2(3t)+ 9cos^2(3t)= 4sin^2(3t)+ 4cos^2(3t)+ 5cos^2(3t)= 4+ 5cos^2(3t)$. Let $u= 4+ 5cos^2(3t)$.

## 1. What is the length of a curve?

The length of a curve is the distance between two points on the curve. It is calculated by finding the sum of infinitely small line segments, also known as infinitesimal elements, that make up the curve.

## 2. How is the length of a curve measured?

The length of a curve is measured using a mathematical concept known as integration. This involves finding the integral of the square root of the sum of the squares of the first derivative of the curve with respect to the independent variable over a given interval.

## 3. What is the formula for finding the length of a curve?

The formula for finding the length of a curve is L = ∫√(1+ (dy/dx)^2) dx, where L is the length of the curve, dy/dx is the first derivative of the curve, and the integral is taken over a given interval.

## 4. What is the significance of finding the length of a curve?

Finding the length of a curve has various applications in different fields such as physics, engineering, and mathematics. It is used to calculate the distance traveled by an object along a curved path, determine the surface area of three-dimensional objects, and find the arc length of a circle, among others.

## 5. Are there any limitations to using integration to find the length of a curve?

Yes, there are limitations to using integration to find the length of a curve. It can only be used for continuous curves and may not work for curves with sharp turns or corners. In such cases, other methods such as approximations or numerical techniques may be used.