Chapter 6: Applications of Definite Integrals - Section 6.2 - Volumes Using Cylindrical Shels - Exercises 6.2 - Page 329: 12

$14 \pi $

We need to use the shell model as follows: $V=\int_p^{q} (2 \pi) \cdot (\space radius \space of \space shell) ( height \space of \space Shell) \space dx$ $ \implies V= \int_1^{4} (2 \pi) \cdot (x)[\dfrac{3}{2 \sqrt x}] dx$ Now, $V=3 \pi [\dfrac{x^{3/2}}{3/2}]_{1}^{4}$ or, $=2 \pi \times (8-1)$ or, $=14 \pi $