---
title: "Double integral y over region between two parabolas"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:double-integral-y-over-region"
source: https://www.bctengineer.com/q/semester_2/mathematics/double-integral-y-over-region
site: BCT Engineering
---

# Double integral y over region between two parabolas

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2071 Magh, 5 marks
- Video: [area between two parabolas   by double integrals](https://www.youtube.com/watch?v=W12PQfZ-XYM) — MATHtech-Maths for all
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/double-integral-y-over-region
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Evaluate: ∬_R y dy dx where R is the region bounded by the parabolas y² = 4x and x² = 4y.

## Short answer

The parabolas intersect at (0,0) and (4,4). For 0≤x≤4, y from x²/4 to 2√x. Integrating y dy dx yields I = 48/5.

## Answer

**Region.** Bounded by the parabolas y² = 4x and x² = 4y.

#### Intersection
From x² = 4y, y = x²/4. Put in y² = 4x: x⁴/16 = 4x, so x(x³ − 64) = 0, giving x = 0 and x = 4. Points: (0,0) and (4,4).

For 0 ≤ x ≤ 4, the upper curve is y = 2√x (from y² = 4x) and the lower curve is y = x²/4.

```text
y
4 |        .*(4,4)
  |     . ' /
  |   .  ' /   y² = 4x (upper)
  |  . '  /    x² = 4y (lower)
  | .'__/______ x
 0           4
```

#### Integral
```text
I = ∫₀⁴ ∫_{x²/4}^{2√x} y dy dx
  = ∫₀⁴ (1/2) [ 4x − x⁴/16 ] dx
  = (1/2) [ 2x² − x⁵/80 ]₀⁴
  = (1/2) ( 32 − 1024/80 )
  = (1/2)(32 − 12.8)
  = 9.6 = 48/5
```

**Answer: I = 48/5.**

## Check yourself

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

1. Parabolas y²=4x, x²=4y meet at x:
   A. 4 only
   B. 0 and 16
   C. 0 and 2
   D. 0 and 4
   Answer: D. x⁴/16=4x. (Key point: Intersection)

2. For 0≤x≤4 upper curve:
   A. y = x²/4
   B. y = x
   C. y = 2√x
   D. y = 4
   Answer: C. y²=4x. (Key point: Upper curve)

3. Value:
   A. 16/3
   B. 9.6/2
   C. 32/5
   D. 48/5
   Answer: D. (1/2)(32−12.8)=9.6=48/5. (Key point: Result)

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