Arc Length

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 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

When y = f(x), when x= g(y), It's basically the same thing, but g'(y) is squared instead

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