---
title: "Double integral r³ between two circles"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:double-integral-r3-between-two"
source: https://www.bctengineer.com/q/semester_2/mathematics/double-integral-r3-between-two
site: BCT Engineering
---

# Double integral r³ between two circles

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2069 Poush, 5 marks
- Video: [Double Integral Problem between two Circles⭕/ Double Integral of f(r, theta)/ multiple Integral](https://www.youtube.com/watch?v=eGqQS3fPUfw) — TEAM Education
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/double-integral-r3-between-two
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate: ∬ r³ dr dθ over the area included between the circles r = 2 sin θ and r = 4 sin θ.

## Short answer

The region lies between the circles r=2 sinθ and r=4 sinθ for θ∈[0,π]. Integrating r³ dr dθ yields 45π/2.

## Answer

**Region.** r = 2 sin θ and r = 4 sin θ are circles through the origin, with diameters 2 and 4 along the y-axis (centres (0,1) and (0,2)). The area between them is the region 2 sin θ ≤ r ≤ 4 sin θ, and θ runs from 0 to π (the circles start and end at the origin).

```text
y
4 |    ___
  |  /  _  \
2 | |  / \  |   inner: r = 2 sinθ
  |  \ \_/ /    outer: r = 4 sinθ
  |____'-'____ x
```

#### Integral
```text
I = ∫₀^π ∫_{2sinθ}^{4sinθ} r³ dr dθ
  = ∫₀^π [ r⁴/4 ]_{2sinθ}^{4sinθ} dθ
  = ∫₀^π (256 − 16)/4 · sin⁴θ dθ
  = 60 ∫₀^π sin⁴θ dθ
```

Using sin⁴θ = (3 − 4 cos 2θ + cos 4θ)/8, ∫₀^π sin⁴θ dθ = 3π/8.

```text
I = 60 · 3π/8 = 45π/2
```

**Answer: 45π/2.**

## Check yourself

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

1. θ ranges:
   A. 0 to π/2
   B. 0 to 2π
   C. 0 to π
   D. π/2 to π
   Answer: C. Circles through origin. (Key point: Limits)

2. Inner integral gives:
   A. 60 sin⁴θ
   B. 16 sin⁴θ
   C. 60 sin²θ
   D. 240 sin⁴θ
   Answer: A. (256−16)/4. (Key point: Inner)

3. Value:
   A. 15π/2
   B. 45π
   C. 60π
   D. 45π/2
   Answer: D. 60·3π/8. (Key point: Result)

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