---
title: "Triple integral of e^(x+y+z)"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:triple-integral-e-x-y"
source: https://www.bctengineer.com/q/semester_2/mathematics/triple-integral-e-x-y
site: BCT Engineering
---

# Triple integral of e^(x+y+z)

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2070 Bhadra, 5 marks
- Video: [Triple Integral of e^(x + y + z) with Logarithmic Limits | Multiple Integrals](https://www.youtube.com/watch?v=cgtML5kCJc0) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/triple-integral-e-x-y
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate ∫₀^(log 2) ∫₀^x ∫₀^(x+log y) e^(x+y+z) dz dy dx.

## Short answer

Integrating e^(x+y+z) in order z, y, x over the given limits gives I = (8/3) ln 2 - 19/9 ≈ -0.263.

## Answer

I = ∫₀^{ln 2} ∫₀^x ∫₀^{x + ln y} e^(x+y+z) dz dy dx

(The upper limit of z is x + ln y. I evaluate the integral exactly as printed.)

#### z-integration
```text
∫₀^{x+ln y} e^(x+y) e^z dz = e^(x+y) [ e^(x + ln y) − 1 ]
                           = e^(x+y) ( y eˣ − 1 )
                           = y e^(2x+y) − e^(x+y)
```

#### y-integration (from 0 to x)
```text
∫₀ˣ y e^(2x+y) dy = e^(2x) [ (y − 1)eʸ ]₀ˣ = e^(2x) [ (x − 1)eˣ + 1 ]
∫₀ˣ e^(x+y) dy    = eˣ (eˣ − 1) = e^(2x) − eˣ
```

Subtract:

```text
(x − 1)e^(3x) + e^(2x) − e^(2x) + eˣ = (x − 1)e^(3x) + eˣ
```

#### x-integration (from 0 to ln 2)
Using ∫ (x − 1)e^(3x) dx = (x − 1)e^(3x)/3 − e^(3x)/9, with e^(3 ln 2) = 8:

```text
[(x − 1)e^(3x)/3 − e^(3x)/9] at ln 2 = 8(ln 2 − 1)/3 − 8/9
                             at 0    = −1/3 − 1/9 = −4/9
difference = 8 ln 2/3 − 8/3 − 8/9 + 4/9 = 8 ln 2/3 − 28/9

∫₀^{ln 2} eˣ dx = 1
```

```text
I = 8 ln 2/3 − 28/9 + 1 = 8 ln 2/3 − 19/9
```

**Answer: I = (8/3) ln 2 − 19/9 ≈ −0.263.**

(The value is negative because for small y the upper limit x + ln y is negative, so the z-integral runs backwards there.)

## Check yourself

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

1. ∫ e^(x+y+z)dz upper x+ln y gives:
   A. y e^x
   B. e^(x+y)(y eˣ −1)
   C. e^(x+y)
   D. e^(2x+y)
   Answer: B. e^(x+ln y)=y eˣ. (Key point: z-integration)

2. Why negative value?
   A. dz negative
   B. e is negative
   C. region is empty
   D. For small y, x+ln y is negative so z-integral runs backwards
   Answer: D. As stated. (Key point: Sign)

3. Value:
   A. (8/3)ln 2
   B. 19/9
   C. (8/3)ln 2 − 19/9
   D. ln 2 − 1
   Answer: C. Final computation. (Key point: Result)

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