How do you find the length of a curve in calculus?

Injenueengado1zy8

Injenueengado1zy8

Answered question

2023-03-24

How to find the length of a curve in calculus?

Answer & Explanation

marallocajcyb

marallocajcyb

Beginner2023-03-25Added 6 answers

In Cartesian coordinates for y = f(x) defined on interval [a,b] the length of the curve is
L = a b 1 + ( d y d x ) 2 d x
In general, we could just write:
L = a b d s Solution:
Let's use Cartesian coordinates for this explanation.
If we consider an arbitrary curve defined as y = f ( x ) and are interested in the interval x [ a , b ] , we can approximate the length of the curve using very tiny line segments.
Consider a point on the curve P i . We can compute the distance of a line segment by finding the difference between two consecutive points on the line | P i - P i - 1 | for i [ 1 , n ] where n is the number of points we've defined on the curve.
This means that the approximate total length of curve is simply a sum of all of these line segments:
L i = 1 n | P i - P i - 1 |
If we want the exact length of the curve, then we can make the assumption that all of the points are infinitesimally separated. We now take the limit of our sum as n .
L = lim n i = 1 n | P i - P i - 1 |
Since we are working in the xy-plane, we can redefine our distance between points to take on the typical definition of Euclidean distance.
| P i - P i - 1 | = ( y i - y i - 1 ) 2 + ( x i - x i - 1 ) 2 = δ y 2 + δ x 2
We can now apply the Mean Value Theorem, which states there exists a point x i lying in the interval [ x i - 1 , x i ] such that
f ( x i ) - f ( x i - 1 ) = f ( x i ) ( x i - x i - 1 )
which we could also write (in the same notation) as
δ y = f ( x i ) δ x
This means that we now have
| P i - P i - 1 | = [ f ( x i ) δ x ] 2 + δ x 2
Simplifying this expression a bit gives us
| P i - P i - 1 | = [ f ( x i ) ] 2 δ x 2 + δ x 2
| P i - P i - 1 | = ( [ f ( x i ) ] 2 + 1 ) δ x 2
| P i - P i - 1 | = ( 1 + [ f ( x i ) ] 2 ) δ x
We can now use this new distance definition in our summation for our points.
L = lim n i = 1 n ( 1 + [ f ( x i ) ] 2 ) δ x
Sums are nice, but integrals are better in continuous situations! Because integrals and sums are both "summation" tools, it's simple to write this as a definite integral. We can also remove our sum index from the integral.
L = a b ( 1 + [ f ( x ) ] 2 ) δ x
Writing this a little bit more typically yields
L = a b ( 1 + ( d y d x ) 2 ) d x
We have arrived at our result! In general, the length is usually defined for a differential of arclength ds
L = a b d s
where ds is defined accordingly for whatever type of coordinate system you are working in. However, I wanted the explanation to be clearer, so I just chose Cartesian ones for simplicity. You could also use polar or spherical coordinates by simply making the necessary substitutions.

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