Find 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 (disk) method.

easy 2 marks Class 12 · Mathematics · Application of Integrals

Answer

\frac{π}{2} m^{3}

The correct answer is $\frac{π}{2} m^{3}$ — Thus the required volume is $\frac{π}{2} m^{3}$ .

Step-by-step solution

The radius of a typical washer is the $y$ ‑coordinate of the curve, i.e. $r = x$ .
Area of the washer is $π r^{2} = π (x)^{2} = π x$ .
Integrate the area from $x = 0$ to $x = 1$ : $V = \int_{0}^{1} π x d x$ .
Evaluate the integral: $V = π [\frac{x ^{2}}{2}]_{0}^{1} = \frac{π}{2}$ .
Thus the required volume is $\frac{π}{2} m^{3}$ .

Common mistake: Forgetting to square the radius, leading to

π x

instead of

π x

Concept tested: Volume by washer method