---
title: "Volume of cardioid r=a(1−cosθ) about initial line"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:volume-cardioid-1-minus-cos"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/volume-cardioid-1-minus-cos
site: BCT Engineering
---

# Volume of cardioid r=a(1−cosθ) about initial line

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2071 Shrawan, 5 marks
- Video: [Find the volume generated by the revolution of the cardioide r=a(1+cosθ) about the initial line](https://www.youtube.com/watch?v=-kc6Nk0OyTA) — EASY MATHS EASY TRICKS
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/volume-cardioid-1-minus-cos
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Find the volume of the solid generated by the revolution of the cardioid r = a(1−cosθ) about the initial line.

## Short answer

Use V = (2π/3)∫ r³ sinθ dθ. With r = a(1-cosθ), substitute u = 1-cosθ to get (2πa³/3)∫₀² u³ du = 8πa³/3 cubic units.

## Answer

For a curve r = f(θ) revolved about the initial line, V = (2π/3) ∫ r³ sin θ dθ.

Cardioid: r = a(1 − cos θ). The upper half (0 ≤ θ ≤ π) generates the full solid.

```text
V = (2π/3) ∫₀^π a³(1 − cosθ)³ sinθ dθ
```
Put u = 1 − cos θ, du = sin θ dθ. Limits: θ = 0 → u = 0; θ = π → u = 2.

```text
V = (2πa³/3) ∫₀² u³ du
  = (2πa³/3) · [u⁴/4]₀²
  = (2πa³/3) · 4
```
**V = 8πa³/3 cubic units.**

## Check yourself

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

1. Volume about the initial line is
   A. π integral r^2 dθ
   B. (2π/3) integral r^3 sin θ dθ
   C. (2π/3) integral r^3 dθ
   D. (π/3) integral r^2 sin θ dθ
   Answer: B. Standard polar formula. (Key point: Formula)

2. With u = 1 - cos θ the limits become
   A. 2 to 0
   B. 0 to 1
   C. 0 to 2
   D. 0 to π
   Answer: C. θ=0 → 0, θ=π → 2. (Key point: Substitution)

3. The integral reduces to
   A. integral u^2 du
   B. integral u^4 du
   C. integral u du
   D. integral u^3 du
   Answer: D. du = sin θ dθ. (Key point: Integrand)

4. V is
   A. 8πa^3/3
   B. 4πa^3/3
   C. 8πa^3
   D. πa^3/3
   Answer: A. (2πa^3/3)·4. (Key point: Result)

## Close variants set in other exams

- [Volume of cardioid r=a(1+cosθ) about initial line](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Avolume-cardioid-1-plus-cos-initial-line) — Mathematics I, chapter 2, asked 6×

## Related in Mathematics I

- [Surface area of cardioid by revolution](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Asurface-area-cardioid-revolution) — Mathematics I, chapter 2