Find the volume of the solid obtained by rotating the region bounded by $y = x$ , $x = 4$ , and the $x$ ‑axis about the $y$ ‑axis using the shell method.

easy 2 marks Class 12 · Mathematics · Application of Integrals

Answer

, \frac{128 π}{5},

The correct answer is $, \frac{128 π}{5},$ — Multiply by the factor $2 π$ : $V = 2 p i c d o t \frac{64}{5} = \frac{128 π}{5}$ .

Step-by-step solution

Set up the shell integral: $V = 2 p iin t_{0}^{4} x s q r t x, d x$ .
Simplify the integrand: $x s q r t x = x^{3/2}$ .
Integrate: $\int_{0}^{4} x^{3/2} d x = \frac{2}{5} x^{5/2}_{0}^{4}$ .
Evaluate the bounds: $\frac{2}{5} (4)^{5/2} = \frac{2}{5} \cdot 32 = \frac{64}{5}$ .
Multiply by the factor $2 π$ : $V = 2 p i c d o t \frac{64}{5} = \frac{128 π}{5}$ .

Common mistake: Forgetting the factor

2 p i

or using incorrect limits

Concept tested: Volume by shell method

Published 7 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked