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

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

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2074 Asoj; 2073 Chaitra; 2072 Kartik; 2069 Chaitra; 2068 Chaitra (asked 5 times)
- 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-plus-cos-initial-line
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

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

## Short answer

Volume V = (2π/3) ∫₀^π r³ sinθ dθ. For r = a(1+cosθ), let u = 1+cosθ. Then V = (2πa³/3) ∫₀² u³ du = (2πa³/3)·4 = 8πa³/3.

## Answer

For a curve r = f(θ) rotated about the initial line (θ = 0), the volume is

V = (2π/3) ∫ r³ sin θ dθ.

For the cardioid r = a(1 + cos θ), the upper half 0 ≤ θ ≤ π generates the whole 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 gives u = 2; θ = π gives u = 0.

```text
V = (2π a³/3) ∫₂⁰ u³ (−du)
  = (2π a³/3) ∫₀² u³ du
  = (2π a³/3) · [u⁴/4]₀²
  = (2π a³/3) · 4
  = 8π a³/3
```
**Answer: V = 8πa³/3 cubic units.**

## Check yourself

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

1. Volume of revolution of r = f(θ) about the initial line is
   A. π integral r^2 dθ
   B. (2π/3) integral r^3 cos θ dθ
   C. (π/3) integral r^2 sin θ dθ
   D. (2π/3) integral r^3 sin θ dθ
   Answer: D. This is the standard polar volume formula. (Key point: Formula)

2. For the cardioid, the whole solid is generated by
   A. θ from 0 to π only
   B. θ from 0 to 2π
   C. θ from 0 to π/2
   D. θ from -π to 0 only
   Answer: A. The upper half already sweeps out the full solid. (Key point: Limits)

3. With u = 1 + cos θ, the limits θ = 0 to π become
   A. u = 0 to u = 2
   B. u = 2 to u = 0
   C. u = 1 to u = 0
   D. u = 2 to u = 1
   Answer: B. Then du = -sin θ dθ flips the limits, giving integral of u^3 from 0 to 2. (Key point: Substitution)

4. The volume is
   A. 4πa^3/3
   B. 8πa^3
   C. 8πa^3/3
   D. 2πa^3/3
   Answer: C. (2πa^3/3)(2^4/4) = 8πa^3/3. (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-minus-cos) — 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