---
title: "Change order of integration: cos y/√((a−x)(a−y))"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:change-order-integration-cos-y"
source: https://www.bctengineer.com/q/semester_2/mathematics/change-order-integration-cos-y
site: BCT Engineering
---

# Change order of integration: cos y/√((a−x)(a−y))

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2068 Bhadra, 5 marks
- Video: [Change of order of Integration in Hindi | double integral | Multiple integral | part 17|Barun maths](https://www.youtube.com/watch?v=QlM-m2bYUAM) — Dr Barun Kumar yadav
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/change-order-integration-cos-y
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Change the order of integration and hence evaluate the same: ∫₀ᵃ ∫₀ˣ cos y dy dx/√((a−x)(a−y)).

## Short answer

Changing the order: y from 0 to a, x from y to a. The inner integral becomes 2√(a-y), cancelling the square root, leaving 2∫₀ᵃ cos y dy = 2 sin a.

## Answer

I = ∫₀^a ∫₀^x cos y dy dx / √((a − x)(a − y))

#### Region
For the given order, y runs from 0 to x and x from 0 to a: the triangle 0 ≤ y ≤ x ≤ a (vertices (0,0), (a,0), (a,a)).

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

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

#### Reversed integral
```text
I = ∫₀^a [ cos y / √(a − y) ] [ ∫_y^a dx / √(a − x) ] dy
```

Inner integral: ∫_y^a (a − x)^(−1/2) dx = [ −2√(a − x) ]_y^a = 2√(a − y).

```text
I = ∫₀^a [ cos y / √(a − y) ] · 2√(a − y) dy
  = 2 ∫₀^a cos y dy
  = 2 sin a
```

Changing the order removes the awkward square roots, which is why it helps.

**Answer: I = 2 sin a.**

## Check yourself

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

1. What region is described by 0 <= y <= x, 0 <= x <= a?
   A. The square 0 <= x, y <= a
   B. The triangle with vertices (0,0), (a,0), (a,a)
   C. The triangle with vertices (0,0), (0,a), (a,a)
   D. The triangle with vertices (0,0), (a,0), (0,a)
   Answer: B. y <= x lies below the line y = x, and x runs to a, giving the triangle (0,0), (a,0), (a,a). (Key point: Region of integration)

2. After changing the order, what are the limits?
   A. x from 0 to a; for each x, y from x to a
   B. y from 0 to a; for each y, x from 0 to y
   C. y from 0 to a; for each y, x from y to a
   D. y from 0 to a; for each y, x from y to 0
   Answer: C. A horizontal strip at height y runs from the line x = y to x = a. (Key point: New limits)

3. What is the integral of 1/sqrt(a - x) dx from x = y to x = a?
   A. 2 sqrt(a - y)
   B. sqrt(a - y)
   C. -2 sqrt(a - y)
   D. 2 sqrt(a) - 2 sqrt(y)
   Answer: A. The antiderivative is -2 sqrt(a - x). At x = a it is 0 and at x = y it is -2 sqrt(a - y), so the result is 2 sqrt(a - y). (Key point: Inner integral)

4. What is I, and why does reversing the order help?
   A. I = 2 sin a, because sqrt(a - y) cancels leaving 2 cos y
   B. I = sin a, because the limits become constants
   C. I = 2 cos a, because sqrt(a - y) cancels leaving 2 sin y
   D. I = 2 sin a, because cos y integrates to a constant
   Answer: A. The factor 2 sqrt(a - y) cancels 1/sqrt(a - y), leaving 2 times the integral of cos y from 0 to a, which is 2 sin a. (Key point: Result and benefit)

## Related in Mathematics

- [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
- [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×
- [Change order: x/(x²+y²)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Achange-order-x-x2-y2) — Mathematics, chapter 2, asked 3×
- [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

- [Solve sin x dy/dx + y cos x = x sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-sinx-dy-dx-y-cosx) — Mathematics I, chapter 4
- [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
- [Solve (1+y²)+(x−e^(tan⁻¹y))dy/dx=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-1-plus-y2-x-minus-e-atan-y) — Mathematics I, chapter 4, asked 6×
- [Solve dy/dx = y/x + tan(y/x)](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-dy-dx-y-over-x-tan) — Mathematics I, chapter 4, asked 6×