---
title: "Change order: x/(x²+y²)"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:change-order-x-x2-y2"
source: https://www.bctengineer.com/q/semester_2/mathematics/change-order-x-x2-y2
site: BCT Engineering
---

# Change order: x/(x²+y²)

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2074 Bhadra, 5 marks
- Video: [Change of order of integration(problem 4)](https://www.youtube.com/watch?v=xvz3vTHfws8) — MATH time
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/change-order-x-x2-y2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate ∫₀ᵃ ∫_y^a x dx dy/(x²+y²) by changing order of integration. [Limits as printed: outer 0 to a, inner y to a; scan unclear]

## Short answer

Reverse the order to x from 0 to a, y from 0 to x. The integral becomes ∫₀ᵃ ∫₀ˣ x dy dx/(x²+y²) = ∫₀ᵃ (π/4) dx = πa/4.

## Answer

**How I read the limits.** As printed: I = ∫₀^a ∫_y^a x dx dy/(x² + y²), with y the outer variable (0 to a) and x the inner variable (from y to a).

#### Region
0 ≤ y ≤ a and y ≤ x ≤ a. This is the triangle with vertices (0,0), (a,0), (a,a), lying between y = 0, y = x and x = a.

```text
y
a |       *(a,a)
  |     / |
  |   /   |
  | /_____|__ x
 0        a
```

Reversing the order: x runs from 0 to a, and for each x, y runs from 0 to x.

#### Reversed integral
```text
I = ∫₀^a x [ ∫₀^x dy/(x² + y²) ] dx
```

Inner: ∫₀^x dy/(x² + y²) = (1/x) tan⁻¹(y/x) |₀^x = (1/x)(π/4).

```text
I = ∫₀^a x · (π/4x) dx = (π/4) ∫₀^a dx = π a/4
```

**Answer: π a/4.**

## Check yourself

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

1. Triangle with vertices:
   A. (0,0),(a,0),(a,a)
   B. (0,0),(0,a),(a,a)
   C. (0,0),(a,0),(0,a)
   D. (a,0),(a,a),(0,a)
   Answer: A. y≤x≤a. (Key point: Region)

2. Reversed order:
   A. y 0 to a, x 0 to y
   B. x 0 to a, y x to a
   C. x 0 to y
   D. x 0 to a, y 0 to x
   Answer: D. For each x, y from 0 to x. (Key point: Reversed limits)

3. Value:
   A. πa/4
   B. πa²/4
   C. πa/2
   D. a/4
   Answer: A. (π/4)∫dx. (Key point: Result)

## Close variants set in other exams

- [Change order: y²/√(y⁴−a²x²)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Achange-order-y2-sqrt-y) — Mathematics, chapter 2, asked 3×

## Related in Mathematics

- [Change order of integration: cos y/√((a−x)(a−y))](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Achange-order-integration-cos-y) — Mathematics, chapter 2
- [Evaluate ∫∫ e^(y/x) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-y-x-dy) — Mathematics, chapter 2
- [Evaluate ∫∫ e^(x²) dx dy](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-x2-dx-dy) — Mathematics, chapter 2
- [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×

## Related in other subjects

- [VBT for [Fe(CN)₆]³⁻ inner/outer orbital](https://www.bctengineer.com/question.md?key=semester_2%2Fengineering_chemistry%3Avbt-fecn6) — Engineering Chemistry, chapter 6
- [Inner/outer orbital complexes; VBT of [Fe(CN)₆]³⁻](https://www.bctengineer.com/question.md?key=semester_2%2Fengineering_chemistry%3Ainner-outer-vbt-fecn6) — Engineering Chemistry, chapter 6
- [Actual grade on inner and outer lanes](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-64p-grade-inner-outer) — Transportation Engineering I, chapter 3
- [RK-4: dy/dx = (1+x²)y](https://www.bctengineer.com/question.md?key=semester_4%2Fnumerical_method%3Ark4-y-1-x2) — Numerical Method, chapter 6