According to Table 5.5.1, the "formula" $2 \mathrm{\π} {\int }_a^b\mathrm{\ρ} \mathit{\ⅆ}s$ becomes

An antiderivative for $x \sqrt{1\+4 x^2}$ can be found, for example, by making the elementary substitution $u\=1\+4 x^2$ so that $\mathrm{du}\=8 x \mathrm{dx}$. Then, $x \mathrm{dx}\=\mathrm{du}\/8$ and the indefinite integral becomes

$\int \frac{u^{1\/2}}{8} \mathit{\ⅆ}u\=\frac{1}{8}\frac{u^{3\/2}}{3\/2}\=\frac{{\left(1\+4 x^2\right)}^{3\/2}}{12}$ 

In Figure 5.5.3(a) the  tutor has been applied to the graph of $y\=x^2$ rotated about the $y$-axis. The Plot Options section has been used to impose one-to-one scaling, and the frame axis style. An exact representation of the value of the surface-area integral is returned, along with a floating-point equivalent. Note the SurfaceOfRevolution command at the bottom of the tutor. This command can return the graphs shown in Figures 5.5.3(a-b), the unevaluated integral, or the value of the surface area.

Figure 5.5.3(b) shows the surface of revolution segmented into three frustums, an image that can be generated in the tutor by selecting "Frustums" in the Display section, and then pressing the Display button. The default number of subintervals is 6, but this has been changed to 3 in the figure. In addition to the graph in Figure 5.5.3(b), the tutor also provides the following midpoint Riemann approximating sum.

It takes a considerably greater number of segments for the midpoint Riemann sum to give a better approximation to the actual value of the surface area.