Determine the coordinates $(\bar{x},\bar{y})$ of the centroid of the region bounded by $y=x^3$ and $y=x^2$ between $x=0$ and $x=1$. Express your answer in exact form.

Step-by-step solution

1. Step 1: Compute area $A={\int }_0^1(x^2-x^3)dx={\left[\frac{x^3}{3}-\frac{x^4}{4}\right]}_0^1=\frac{1}{12}$.
2. Step 2: Find $\bar{x}=(1/A){\int }_0^{1}x(x^2-x^3)dx=(1/A){\int }_0^1(x^3-x^4)dx=(1/A)(\frac{1}{4}-\frac{1}{5})=\frac{12}{20}=\frac{3}{5}$.
3. Step 3: Find $\bar{y}=(1/(2A)){\int }_0^1[(x^2{)}^2-(x^3{)}^2]dx=(1/(2A)){\int }_0^1(x^4-x^6)dx=(1/(2A))(\frac{1}{5}-\frac{1}{7})=\frac{6}{1}\left(\frac{2}{35}\right)=\frac{12}{35}$.
4. Step 4: State the centroid coordinates as $(\frac{3}{5},\frac{12}{35})$.

Concept tested: Centroid of region between curves

Published 1 October 2026 · How this solution was prepared and checked