University Calculus: Early Transcendentals (3rd Edition)

$$\dfrac{14 \pi }{3}$$
Consider the shell model to compute the volume: $$V=\int_{m}^{n} (2 \pi) (\space Radius \space of \space shell) \times ( height \space \text{of} \space \text {shell}) dx \\ = \int_0^{\sqrt 3} (2 \pi) \cdot (x)[\sqrt {x^2+1}) dx$$ Let us consider $a=x^2+1$ and $da=2xdx$ $$Volume= \pi \times \int_1^{4} \sqrt a da \\ =\pi \times [\dfrac{2}{3})a^{5/2}]_1^4 \\=\dfrac{14 \pi }{3}$$