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

# Polar coordinates: e^−(x²+y²) integral

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2072 Ashwin, 5 marks; 2068 Magh, 5 marks (asked 2 times)
- Video: [e^{-(x² + y²)} | Polar Coordinates Solve What Cartesian Cannot](https://www.youtube.com/watch?v=sITf-KB975I) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/polar-coordinates-e-x2-y2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate: ∫₀^∞ ∫₀^∞ e^−(x²+y²) dy dx by changing to polar coordinates.

## Short answer

First quadrant, polar: 0≤r<∞, 0≤θ≤π/2. e^{−(x²+y²)}=e^{−r²}, dx dy=r dr dθ. ∫₀^∞ e^{−r²} r dr = 1/2, θ-integral π/2, giving π/4.

## Answer

**Region.** x and y both run from 0 to ∞: the whole first quadrant. In polar coordinates: 0 ≤ r < ∞, 0 ≤ θ ≤ π/2.

#### Change of variables
x² + y² = r², dy dx = r dr dθ.

```text
I = ∫₀^{π/2} ∫₀^∞ e^(−r²) r dr dθ
```

#### Integrate in r
Put t = r², dt = 2r dr:

```text
∫₀^∞ e^(−r²) r dr = (1/2) ∫₀^∞ e^(−t) dt = (1/2)[ −e^(−t) ]₀^∞ = 1/2
```

#### Integrate in θ
```text
I = (1/2) ∫₀^{π/2} dθ = (1/2)(π/2) = π/4
```

**Answer: I = π/4.**

**Consequence.** The integral separates as (∫₀^∞ e^(−x²) dx)² = π/4, so ∫₀^∞ e^(−x²) dx = √π/2.

## Check yourself

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

1. The first quadrant gives:
   A. 0≤r≤1
   B. 0≤r<∞, 0≤θ≤π/2
   C. 0≤θ≤2π
   D. 0≤θ≤π
   Answer: B. Quadrant. (Key point: Limits)

2. ∫₀^∞ e^(−r²) r dr equals:
   A. 2
   B. 1/2
   C. 1
   D. π/2
   Answer: B. Put t = r². (Key point: r-integral)

3. The integral equals π/4, so ∫₀^∞ e^(−x²)dx is:
   A. π/4
   B. π/2
   C. √π/2
   D. √π
   Answer: C. Square root of π/4. (Key point: Result)

## Close variants set in other exams

- [Polar coordinates: y²√(x²+y²)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Apolar-coordinates-y2sqrt-x2-y2) — Mathematics, chapter 2, asked 7×

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