---
title: "Area between x²=4y and x=4y−2"
subject: "Mathematics I"
chapter: 2
key: "semester_1/mathematics_i:area-parabola-and-line"
source: https://www.bctengineer.com/q/semester_1/mathematics_i/area-parabola-and-line
site: BCT Engineering
---

# Area between x²=4y and x=4y−2

Mathematics I · Chapter 2: Integration and its Applications

- Asked in: 2071 Shrawan, 5 marks
- Video: [Area between x^2=4y and x=4y-2||Area between Parabola and Straight Line||Area Under Curve||Calculus](https://www.youtube.com/watch?v=Af-wSMN4IPA) — Tutor - Praveen Singh
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/mathematics_i/area-parabola-and-line
- Whole chapter: https://www.bctengineer.com/md/semester_1/mathematics_i/chapter-2

## Question

Find the area bounded by the curve x² = 4y and the line x = 4y − 2.

## Short answer

Intersection points are at x = -1 and x = 2, with the line above the parabola. The area is ∫_{-1}^{2} [(x+2)/4 - x²/4] dx = 9/8 square units.

## Answer

Curves: parabola x² = 4y and line x = 4y − 2, i.e. y = (x + 2)/4.

**Intersection.** Substitute 4y = x² into x = 4y − 2: x = x² − 2, so x² − x − 2 = 0, (x − 2)(x + 1) = 0. Hence x = 2 or x = −1, giving the points (−1, 1/4) and (2, 1).

Between x = −1 and x = 2 the line lies above the parabola (e.g. at x = 0, line y = 1/2, parabola y = 0).

```text
A = ∫_(−1)^2 [ (x + 2)/4 − x²/4 ] dx
  = (1/4) [ x²/2 + 2x − x³/3 ]_(−1)^2
  = (1/4) [ (2 + 4 − 8/3) − (1/2 − 2 + 1/3) ]
  = (1/4) [ 10/3 + 7/6 ]
  = (1/4)(27/6)
```
**Area = 9/8 square units.**

## Check yourself

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

1. Substituting 4y = x^2 into x = 4y - 2 gives x = 
   A. 2 and 1
   B. 2 and -1
   C. -2 and 1
   D. 0 and 2
   Answer: B. x^2 - x - 2 = 0. (Key point: Intersection)

2. Between x = -1 and 2 the upper curve is
   A. the parabola
   B. neither
   C. the line
   D. both equal
   Answer: C. At x=0, line y = 1/2, parabola y = 0. (Key point: Upper curve)

3. The area is
   A. integral from -1 to 2 of [x^2/4 - (x+2)/4] dx
   B. integral from 0 to 2 of ...
   C. integral from -1 to 2 of (x+2)/4 dx
   D. integral from -1 to 2 of [(x+2)/4 - x^2/4] dx
   Answer: D. Upper minus lower. (Key point: Integral)

4. The area is
   A. 9/8
   B. 9/4
   C. 27/8
   D. 3/8
   Answer: A. (1/4)(27/6) = 9/8. (Key point: Result)

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