---
title: "Area under one arch of a cycloid"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:area-cycloid-arc-and-base"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/area-cycloid-arc-and-base
site: BCT Engineering
---

# Area under one arch of a cycloid

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2073 Chaitra
- Video: [Calculus 2: Parametric Equations (18 of 20) Find the Area of an Arch of a Cycloid](https://www.youtube.com/watch?v=MYZ7ELYn86w) — Michel van Biezen
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/area-cycloid-arc-and-base
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Find the area included between an arc of the cycloid x = a(θ − sinθ), y = a(1 − cosθ) and its base.

## Short answer

Integrate y dx from θ=0 to 2π: a²∫₀²π (1−cosθ)² dθ = 16a²∫₀^{π/2} sin⁴u du = 3πa² square units.

## Answer

Cycloid: x = a(θ − sin θ), y = a(1 − cos θ). One arch is traced as θ goes from 0 to 2π; its base is the x-axis from x = 0 to x = 2πa.

dx = a(1 − cos θ) dθ.

```text
Area = ∫ y dx
     = ∫₀^(2π) a(1 − cosθ) · a(1 − cosθ) dθ
     = a² ∫₀^(2π) (1 − cosθ)² dθ
     = a² ∫₀^(2π) 4 sin⁴(θ/2) dθ
```
Put θ/2 = u, dθ = 2du, limits 0 to π:

Area = 8a² ∫₀^π sin⁴u du = 16a² ∫₀^(π/2) sin⁴u du = 16a² · 3π/16

**Area = 3πa² square units** (three times the area of the generating circle).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. One arch is traced for θ from
   A. 0 to π
   B. 0 to 2π
   C. 0 to π/2
   D. -π to π/2
   Answer: B. Base is 0 to 2πa. (Key point: Range)

2. Area = integral y dx becomes
   A. a^2 integral (1 - cos θ) dθ
   B. a integral (1-cos θ)^2 dθ
   C. a^2 integral (1 - cos θ)^2 dθ
   D. a^2 integral sin^2 θ dθ
   Answer: C. dx = a(1 - cos θ) dθ. (Key point: Integrand)

3. (1 - cos θ)^2 equals
   A. 2 sin^2(θ/2)
   B. 4 sin^2(θ/2)
   C. sin^4 θ
   D. 4 sin^4(θ/2)
   Answer: D. 1 - cos θ = 2 sin^2(θ/2). (Key point: Identity)

4. The area is
   A. 3πa^2
   B. πa^2
   C. 2πa^2
   D. 6πa^2
   Answer: A. Three times the generating circle. (Key point: Result)

## Related in Mathematics I

- [Volume or surface area of cycloid about base](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-or-surface-cycloid-base) — Mathematics I, chapter 2
- [Area of one loop of r = a sin 3θ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aarea-one-loop-sin-3theta) — Mathematics I, chapter 2
- [DUIS: ∫ dx/(a²sin²x+b²cos²x)²](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aduis-dx-over-a2sin2-b2cos2-squared) — Mathematics I, chapter 2, asked 2×
- [Volume of cycloid about tangent at vertex](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-cycloid-about-tangent-at-vertex) — Mathematics I, chapter 2
- [Beta-Gamma: ∫ cos⁴3θ sin²6θ dθ = 5π/192](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Abeta-gamma-cos4-3theta-sin2-6theta) — Mathematics I, chapter 2, asked 3×
- [Gamma function: ∫ sin⁶(x/2) cos⁶(x/2) dx](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Agamma-sin6-cos6-half-x) — Mathematics I, chapter 2, asked 3×

## Related in other subjects

- [Sampling theorem; Nyquist of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-theorem-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 5×
- [Aliasing; Nyquist rate of 1 + cos + sin](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aaliasing-nyquist-1-cos-sin) — Signals and Systems, chapter 4, asked 4×
- [DTFS of 1 + sin + 3cos + cos(4πn/N + π/2)](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Adtfs-coefficients-with-phase-term) — Signals and Systems, chapter 2, asked 3×
- [Gauss 3-point: sin x³ + cos x²](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Agauss-3pt-sin-x3-cos-x2) — Numerical Method, chapter 5, asked 2×