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:

Have a nice day!

proofs involving isosceles triangles often require special consideration because an isosceles triangle has several distinct properties that do not apply to normal triangles.(more about triangle types) therefore, when you are trying to prove that two triangles are congruent, and one or both triangles, are isosceles you have a few theorems that you can use to make your life easier.

step-by-step explanation:

an isosceles triangle has two congruent sides and two congruent angles. the congruent angles are called the base angles and the other angle is known as the vertex angle. ∠bac and ∠bca are the base angles of the triangle picture on the left. the vertex angle is ∠abc