Find the area under one arch of the cycloid x = r(θ − sin(θ)) y = r(1 − cos(θ

Step-by-step explanation:

We can define the area under arch of the cycloid as:

Let's evaluate this integral between 0 and 2π and put it in terms of dθ, using the chain rule.

(1)

Taking the derivative of x we have:

(2)

Now, we can put (2) in (1).

We can solve the quadratic equation to solve this integral:

Now, we just need to take this integral by the sum rule. Let's recall we can use integration by part to solve cos²(θ)dθ.

Finally, the area is:

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