Compute the volume (in cubic metres) of the solid obtained by rotating the region bounded by $y=x$, the $x$‑axis, and the vertical lines $x=0$ and $x=1$ about the $x$‑axis using the washer method.

Step-by-step solution

1. Step 1: Set up the integral using washers: $V=\pi {\int }_0^1[f(x){]}^{2}dx$, where $f(x)=x$.
2. Step 2: Compute the integrand: $(x{)}^2=x^2$.
3. Step 3: Integrate: ${\int }_0^1{x}^{2}dx={\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$.
4. Step 4: Multiply by $\pi$: $V=\pi \cdot \frac{1}{3}=\frac{\pi }{3}$.
5. Step 5: Conclude the volume is $\frac{\pi }{3}{\text{m}}^3$.

Concept tested: Volume by washer method

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