Arc length problems:
If we were to find the length of a function that is continuous over the interval [a,b] for the function y = f(x), we would call that a problem of finding arc length.
Arc length as limit of sum:
Since we are trying to estimate the length of a curve, we would do that by splitting the curve into ‘n’ parts. The width of each part would be 𝛥X . Each of the points on the curve would be P0,P1,P3,… Pi,…. Pn. So that P0 is the starting point of the curve at x = a and Pn is the ending point of the curve at x = b.
The length between P0 and P1 can be denoted by |P0 – P1|. The length between P1 and P2 would be |P2-P1| so similarly the length between Pi-1 and Pi would be |P(i-1) – Pi|. In such case the approximate length of the curve would be L = ∑_(i=1)^n▒|P(i-1)- Pi| . But here what is n? As n gets larger and larger, the distance |P(i-1) – Pi| gets smaller and smaller and retrospectively the L becomes more accurate. So now we can say that L = lim┬(n→∞)∑_(i=1)^n▒|P(i-1)- Pi| .
The co-ordinates of any point Pi are (xi,yi) and similarly that of P(i-1) are (X(i-1), Y(i-1)). Therefore the distance between P(i-1) and Pi using distance formula would be:
|P(i-1) – Pi| = √[(X(i-1) – xi)^2 + (Y(i-1) – yi)^2].
Here we already established that the distance between two consecutive points on x axis was 𝛥x and let Y(i-1) – Yi = 𝛥yi. Then,
|P(i-1) – Pi| = √[ 𝛥x^2 + 𝛥yi^2]
So that now,
L = lim┬(n→∞)∑_(i=1)^n▒〖√[ Δx^2 + Δyi^2]〗
Arc length integral:
We know that limit of a sum can be written as an integral. Therefore the above limit of sum can be written as an integral as follows:
L = ∫_a^b▒〖√[ Δx^2 + Δyi^2]〗.
Since we are talking of integrals we can replace the 𝛥x and 𝛥yi with dx and dy, so we have:
L = ∫_a^b▒〖√[ dx^2 + dy^2]〗. Factoring out the dx^2 we have:
L = ∫_a^b▒〖√dx^2[1 + dy^2/dx^2]〗. Bringing the dx^2 outside the root house, makes it dx. So that now we have:
L = ∫_a^b▒〖√[1 +dy^2/dx^2 ]dx〗. That is the same as writing,
L = ∫_a^b▒〖√[1 +(dy^ /dx^ )^2]dx〗
This is the formula to be used to calculate arc length of a curve. Alternatively the formula can also be stated as:
L = ∫_a^b▒〖√[1 +(f^' (x))^2]dx〗