---
title: "Prove triple integral of √(x²+y²+z²)e^−(x²+y²+z²) is 2π"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:prove-triple-integral-sqrt-x2"
source: https://www.bctengineer.com/q/semester_2/mathematics/prove-triple-integral-sqrt-x2
site: BCT Engineering
---

# Prove triple integral of √(x²+y²+z²)e^−(x²+y²+z²) is 2π

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2068 Magh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/prove-triple-integral-sqrt-x2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Prove that ∫_(−∞)^∞ ∫_(−∞)^∞ ∫_(−∞)^∞ √(x²+y²+z²) e^−(x²+y²+z²) dx dy dz = 2π. [Integrand as read from scan; the √ factor is unclear]

## Short answer

Switch to spherical coordinates: integrand r·e^{-r²}·r² sinφ dr dφ dθ. Angle integrals give 4π, radial integral ∫₀∞ r³ e^{-r²} dr = 1/2, product is 2π.

## Answer

**Reading.** The integrand is read as √(x² + y² + z²) · e^−(x² + y² + z²), the √ factor being the radius. This matches the stated value 2π.

I = ∭ over all space √(x² + y² + z²) e^−(x²+y²+z²) dx dy dz.

#### Spherical polar coordinates
x = r sin φ cos θ, y = r sin φ sin θ, z = r cos φ, dx dy dz = r² sin φ dr dφ dθ. Since x² + y² + z² = r², the whole space is covered by 0 ≤ r < ∞, 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π.

```text
I = ∫₀^{2π} dθ ∫₀^π sinφ dφ ∫₀^∞ r · e^(−r²) · r² dr
  = (2π)(2) ∫₀^∞ r³ e^(−r²) dr
```

#### The radial integral
Put t = r², dt = 2r dr, so r³ dr = (1/2) t dt:

```text
∫₀^∞ r³ e^(−r²) dr = (1/2) ∫₀^∞ t e^(−t) dt = (1/2) Γ(2) = 1/2
```

#### Result
```text
I = 4π · (1/2) = 2π
```

Hence the integral equals 2π. ∎

## Check yourself

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

1. Why are spherical polar coordinates suitable?
   A. The integrand depends only on x^2 + y^2 + z^2 = r^2 and the region is all space
   B. The integrand is linear in x, y, z
   C. The region is a cylinder
   D. The integrand is independent of r
   Answer: A. Since the integrand depends only on r, spherical coordinates separate the integral. (Key point: Choice of coordinates)

2. What is the product of the angular integrals of d theta and sin(phi) d phi?
   A. 2 pi
   B. pi
   C. 4 pi
   D. 8 pi
   Answer: C. The d theta integral is 2 pi and the sin(phi) integral from 0 to pi is 2, so 4 pi. (Key point: Angular factor)

3. With t = r^2, what does the integral of r^3 e^(-r^2) dr become?
   A. The integral of t e^(-t) dt
   B. (1/2) times the integral of t e^(-t) dt
   C. (1/2) times the integral of t^2 e^(-t) dt
   D. 2 times the integral of t e^(-t) dt
   Answer: B. dt = 2r dr gives r^3 dr = (1/2) t dt. (Key point: Radial substitution)

4. What is the radial integral and the final result?
   A. 1/2 and I = 2 pi
   B. 1 and I = 4 pi
   C. 1/2 and I = 4 pi
   D. Gamma(3) = 2 and I = 8 pi
   Answer: A. (1/2)Gamma(2) = 1/2, and 4 pi x 1/2 = 2 pi. (Key point: Final value)

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