---
title: "Polar coordinates: y²√(x²+y²)"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:polar-coordinates-y2sqrt-x2-y2"
source: https://www.bctengineer.com/q/semester_2/mathematics/polar-coordinates-y2sqrt-x2-y2
site: BCT Engineering
---

# Polar coordinates: y²√(x²+y²)

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2073 Bhadra, 5 marks; 2072 Magh, 5 marks; 2070 Magh, 5 marks (asked 3 times)
- Video: [#6 Evaluation of Double integral by changing into Polar 22MATX21|VTU|SUK|21MAT21|18MAT21](https://www.youtube.com/watch?v=EtymzfXg5CI) — CGAT Maths
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/polar-coordinates-y2sqrt-x2-y2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate the integral by changing to polar coordinates ∫₀ᵃ ∫₀^√(a²−x²) y² √(x²+y²) dy dx.

## Short answer

Region is quater of circle radius a in first quadrent. Change to polar: y²√(x²+y²) dy dx = r⁴ sin²θ dr dθ. Integrate r from 0 to a, θ from 0 to π/2: I = (a⁵/5)(π/4) = π a⁵/20.

## Answer

**Region.** x runs from 0 to a and y from 0 to √(a² − x²): this is the quarter of the circle x² + y² = a² lying in the first quadrant. In polar coordinates: 0 ≤ r ≤ a, 0 ≤ θ ≤ π/2.

#### Change of variables
x = r cos θ, y = r sin θ, √(x² + y²) = r, dy dx = r dr dθ.

```text
y² √(x²+y²) dy dx = (r² sin²θ)(r)(r dr dθ) = r⁴ sin²θ dr dθ
```

#### Integrate
```text
I = ∫₀^{π/2} sin²θ dθ · ∫₀^a r⁴ dr
  = ∫₀^{π/2} sin²θ dθ · (a⁵/5)
```

Now ∫₀^{π/2} sin²θ dθ = ∫₀^{π/2} (1 − cos 2θ)/2 dθ = π/4.

```text
I = (π/4)(a⁵/5) = π a⁵/20
```

**Answer: π a⁵/20.**

## Check yourself

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

1. Region of integration:
   A. Square of side a
   B. Quarter disc in first quadrant, 0≤r≤a, 0≤θ≤π/2
   C. Half disc
   D. Full disc
   Answer: B. x 0..a, y 0..√(a²−x²). (Key point: Region)

2. The integrand with area element becomes:
   A. r³ sin²θ dr dθ
   B. r⁴ sin²θ dr dθ
   C. r⁴ sin θ dr dθ
   D. r² sin²θ dr dθ
   Answer: B. (r² sin²θ)(r)(r dr dθ). (Key point: Transformation)

3. The value is:
   A. πa⁵/10
   B. πa⁵/20
   C. a⁵/5
   D. πa⁴/16
   Answer: B. (π/4)(a⁵/5). (Key point: Result)

## Close variants set in other exams

- [Polar coordinates: ∫∫(x²+y²) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-x2-y2-dy) — Mathematics, chapter 2, asked 5×
- [Polar coordinates: e^−(x²+y²) integral](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-e-x2-y2) — Mathematics, chapter 2, asked 5×

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