Chapter 9 - Further Applications of the Integral and Taylor Polynomials - 9.1 Arc Length and Surface Area - Exercises - Page 469: 37

$\approx 203 $

Since \begin{aligned} S&=2 \pi \int_{a}^{b} f(x) \sqrt{1+\left[f'(x)\right]^{2}} d x\\ &=2 \pi \int_{0}^{2} x^{3} \sqrt{1+\left[3 x^{2}\right]^{2}} d x \\ &=2 \pi \int_{0}^{2} x^{3} \sqrt{1+9 x^{4}} d x\\ &=\frac{\pi}{18}\int_{0}^{2}36 x^{3} \sqrt{1+9 x^{4}} d x\\ &= \frac{ \pi}{27}\left(1+9 x^{4}\right)^{3/2}\bigg|_{0}^{2}\\ &\approx 203 \end{aligned}