---
title: "Area bounded by y²=x³ and y=x"
subject: "Mathematics"
chapter: 2
key: "semester_2/mathematics:area-bounded-y2-x3-y"
source: https://www.bctengineer.com/q/semester_2/mathematics/area-bounded-y2-x3-y
site: BCT Engineering
---

# Area bounded by y²=x³ and y=x

Mathematics · Chapter 2: Multiple Integrals

- Asked in: 2071 Bhadra, 5 marks
- Video: [Finding the Area Between Two Curves by Integration](https://www.youtube.com/watch?v=xec6HTcn2M8) — Professor Dave Explains
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/area-bounded-y2-x3-y
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-2

## Question

Find, by double integration, the area of the region bounded by y² = x³ and y = x.

## Short answer

The curves intersect at (0,0) and (1,1). For 0≤x≤1, y runs from x^(3/2) to x. The double integral gives area = ∫₀¹ (x - x^(3/2)) dx = 1/10 square unit.

## Answer

**Curves.** y² = x³ (a cuspidal curve; its upper branch is y = x^(3/2)) and the line y = x.

#### Intersection
Put y = x in y² = x³: x² = x³, so x²(1 − x) = 0, giving x = 0 and x = 1. Points: (0,0) and (1,1).

For 0 < x < 1, x^(3/2) < x, so the curve lies below the line and the region lies between y = x^(3/2) and y = x.

```text
y
1 |           *(1,1)
  |        . /
  |      ./ .      line y = x
  |    ./ .        curve y² = x³
  |  _/ .
  |_/_______ x
 0          1
```

#### Area by double integration
```text
A = ∬ dy dx = ∫₀¹ ∫_{x^{3/2}}^{x} dy dx
  = ∫₀¹ ( x − x^{3/2} ) dx
  = [ x²/2 − (2/5) x^{5/2} ]₀¹
  = 1/2 − 2/5
  = 1/10
```

**Answer: Area = 1/10 square unit.**

## Check yourself

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

1. Intersection x:
   A. −1,1
   B. 0 and 1
   C. 1 only
   D. 0 and 2
   Answer: B. x²=x³. (Key point: Intersection)

2. For 0<x<1, which is lower?
   A. y = x^(3/2)
   B. y = x
   C. Neither; they do not meet
   D. Both curves coincide for 0<x<1
   Answer: A. x^(3/2)<x. (Key point: Which is lower)

3. Area:
   A. 1/6
   B. 1/10
   C. 1/2
   D. 1/5
   Answer: B. 1/2−2/5. (Key point: Area)

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