It's important to find the length of an arc. Let's say you need a curved tarp for a tent, and you have an equation for the tarp, but you want to know its length. Luckily, calculus was invented/discovered.
So, in order to find the length of a line segment, you can keep taking smaller and smaller distance formulas.
Σ√((xi - xi-1)2 + (yi - yi-1)2))
That's the distance formula, you can also write it as:
Σ√((Δx)2 + (Δy)2)
If you multiply the y2 term by x2/x2 you get:
Σ√((Δx)2 + (Δy/Δx)2(Δx)2)
Note that this is mathematically equivalent to the previous expression
Now, we can take out a √((Δx)2), which is just Δx.
So, we get
Σ√(1 + (Δy/Δx)2)Δx
Recall that's just the definition of an integral. Our final answer for the length of an arc length when y = f(x) is
It's similar, but you square the derivatives of both the x and y
Say y = rsin(θ), and x = rcos(θ), (y')^2 + (x')^2 =
(r'sin(θ) + rcos(θ))^2 + (r'cos(θ)- rsin(θ))^2 =
r'^2sin^2(θ) + 2r'rsin(θ)cos(θ) + r^2cos^2(θ) + r'^2cos^2(θ) - 2rr'cos(θ)sin(θ) + r^2sin^2(theta) =
r'^2(sin^2(θ) + cos^2(θ)) + r^2(sin^2(θ) + cos^2(θ)) = r'^2 + r^2, so the equation for polar arc length is