# Find the length of the curve C

• chwala
In summary: So, in summary, the highlighted ± sign in the equation for arc length indicates that the derivative can have either a positive or negative sign, and in this particular problem, the derivative has a negative sign.
chwala
Gold Member
Homework Statement
see attached
Relevant Equations
Integration
This question is from a Further Maths paper;

Part (a) is pretty straight forward...No issue here...one has to use chain rule...

Let ##U=\dfrac{e^x+1}{e^x-1}## to realize ##\dfrac{du}{dx}=\dfrac{-2e^x}{(e^x-1)^2}##

and let
##y=\ln u## on taking derivatives, we shall have ##\dfrac{dy}{du}=\dfrac{e^x-1}{e^x+1}##

therefore,
##\dfrac{dy}{dx}=\dfrac{e^x-1}{e^x+1}×\dfrac{-2e^x}{(e^x-1)^2}=-\dfrac{2e^x}{(e^{2x}-1)^2}##Now my question (reason for posting this problem is on part b). Why do we have ± on the highlighted...i thought we are taking absolute value of ##\dfrac{dy}{dx}## which would be a positive.

i.e Arc length = $$\int_a^b \sqrt{1+\left[f^{'}(x)\right]^2} dx$$

Thanks.

Last edited:
chwala said:
Homework Statement:: see attached
Relevant Equations:: Integration

This question is from a Further Maths paper;

View attachment 315769

Part (a) is pretty straight forward...No issue here...one has to use chain rule...

Let ##U=\dfrac{e^x+1}{e^x-1}## to realize ##\dfrac{du}{dx}=\dfrac{-2e^x}{(e^x-1)^2}##

and let
##y=\ln u## on taking derivatives, we shall have ##\dfrac{dy}{du}=\dfrac{e^x-1}{e^x+1}##

therefore,
##\dfrac{dy}{dx}=\dfrac{e^x-1}{e^x+1}×\dfrac{-2e^x}{(e^x-1)^2}=\dfrac{-2e^x}{(e^{2x}-1)^2}##Now my question (reason for posting this problem is on part b). Why do we have ± on the highlighted...i thought we are taking absolute value of ##\dfrac{dy}{dx}## which would be a positive.

i.e Arc length = $$\int_a^b \sqrt{1+\left[f^{'}(x)\right]^2} dx$$

View attachment 315771Thanks.
Aaaaargh, I've seen the reason... ...the sign on the derivative may either be positive or negative ...in our problem the derivative is having a negative sign. i.e ##\dfrac{dy}{dx}=-\dfrac{2e^x}{(e^{2x}-1)^2}##

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

The length of a curve is the distance along the curve between two points, usually measured in units such as meters or feet.

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

The length of a curve is calculated using a mathematical formula called the arc length formula. This formula takes into account the shape of the curve and the coordinates of its points to determine the length.

## 3. Can the length of a curve be infinite?

Yes, the length of a curve can be infinite if the curve extends infinitely in one or both directions. An example of this is a straight line or a parabola with no defined endpoints.

## 4. Is there a difference between the length of a curve and the distance between its endpoints?

Yes, the length of a curve takes into account the actual shape of the curve, while the distance between its endpoints only measures the straight-line distance between the two points.

## 5. How is finding the length of a curve useful in real-world applications?

Finding the length of a curve is useful in various fields such as engineering, physics, and architecture. It allows for accurate measurements and calculations in designing structures, creating maps, and analyzing motion and forces in objects.

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