# Arc Length

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√((x_{i} - x_{i-1})^{2} + (y_{i} - y_{i-1})^{2}))

That's the distance formula, you can also write it as:

## Σ√((Δx)^{2} + (Δy)^{2})

If you multiply the y^{2} term by x^{2}/x^{2} 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

# Parametric

It's similar, but you square the derivatives of both the x and y

# Polar

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