---
title: "Centroid x of area under y = 12x − 1.5x²"
subject: "Applied Mechanics"
chapter: 4
key: "semester_1/applied_mechanics:centroid-x-coordinate-y-12x-1-5x2"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/centroid-x-coordinate-y-12x-1-5x2
site: BCT Engineering
---

# Centroid x of area under y = 12x − 1.5x²

Applied Mechanics · Chapter 4: Center of Gravity, Centroid and Moment of Inertia

- Asked in: 2069 Chaitra, 4 marks
- Video: [Statics 9.9 and 9.10 - Locate the centroid of the shaded area.](https://www.youtube.com/watch?v=9Jy42jB6Jn8) — Learning by Teaching
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/centroid-x-coordinate-y-12x-1-5x2
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-4

## Question

Determine centroidal x coordinate of the shaded area shown in figure below. [Figure: area under the curve y = 12x − 1.5x² above the x axis, from x = 0 to x = 4 m.]

## Short answer

Using vertical strip, area = 64 m², first moment = 160 m³, so centroidal x-coordinate = 2.5 m.

## Answer

**Setting up:** the area lies under y = 12x − 1.5x² from x = 0 to x = 4 m (y at x = 4 is 24 m). Take a vertical strip of height y and width dx, with x̃ = x.

#### Area
A = ∫₀⁴ (12x − 1.5x²) dx = [6x² − 0.5x³]₀⁴ = 6(16) − 0.5(64) = 96 − 32 = 64 m²

#### First moment about the y axis
∫ x dA = ∫₀⁴ x (12x − 1.5x²) dx = ∫₀⁴ (12x² − 1.5x³) dx = [4x³ − 0.375x⁴]₀⁴
= 4(64) − 0.375(256) = 256 − 96 = 160 m³

#### Centroid
x̄ = 160/64 = **2.5 m** from the y axis.

## Check yourself

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

1. What is the area under y = 12x - 1.5x^2 from 0 to 4 m?
   A. 24 m^2
   B. 32 m^2
   C. 64 m^2
   D. 96 m^2
   Answer: C. [6x^2 - 0.5x^3] from 0 to 4 = 96 - 32 = 64. (Key point: Area)

2. Which integral gives the first moment about the y axis?
   A. integral of y^2 dx
   B. integral of x dx
   C. integral of x y dx
   D. integral of y dx
   Answer: C. x̃ = x for a vertical strip, so integral x dA = integral x y dx. (Key point: Moment formula)

3. x̄ equals:
   A. 2.5 m
   B. 3 m
   C. 1.6 m
   D. 2 m
   Answer: A. 160/64 = 2.5 m. (Key point: Result)

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