Paul's Online Notes
Paul's Online Notes
Home / Calculus II / Applications of Integrals / Surface Area
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 2-2 : Surface Area

4. Find the surface area of the object obtained by rotating \(y = 4 + 3{x^2}\) , \(1 \le x \le 2\) about the \(y\)-axis.

Show All Steps Hide All Steps

Start Solution

The first step here is to decide on a \(ds\) to use for the problem. We can use either one, however the function is set up for,

\[ds = \sqrt {1 + {{\left[ {\frac{{dy}}{{dx}}} \right]}^2}} \,dx\]

Using the other \(ds\) will put fractional exponents into the function and make the \(ds\) and integral potentially messier so we’ll stick with this \(ds\).

Show Step 2

Let’s now set up the \(ds\).

\[\frac{{dy}}{{dx}} = 6x\hspace{0.5in} \Rightarrow \hspace{0.5in}ds = \sqrt {1 + {{\left[ {6x} \right]}^2}} \,dx = \sqrt {1 + 36{x^2}} \,dx\] Show Step 3

The integral for the surface area is,

\[SA = \int_{{}}^{{}}{{2\pi x\,ds}} = \int_{1}^{2}{{2\pi x\,\sqrt {1 + 36{x^2}} \,dx}}\]

Note that because we are rotating the function about the \(y\)-axis for this problem we need an \(x\) in front of the root. Also note that because our choice of \(ds\) puts a \(dx\) in the integral we need \(x\) limits of integration which we were given in the problem statement.

Show Step 4

Finally, all we need to do is evaluate the integral. That requires a quick Calc I substitution. We’ll leave most of the integration details to you to verify since you should be pretty good at Calc I substitutions by this point.

\[SA = \int_{1}^{2}{{2\pi x\,\sqrt {1 + 36{x^2}} \,dx}} = \left. {\frac{\pi }{{54}}{{\left( {1 + 36{x^2}} \right)}^{\frac{3}{2}}}} \right|_1^2 = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{\pi }{{54}}\left( {{{145}^{\frac{3}{2}}} - {{37}^{\frac{3}{2}}}} \right) = 88.4864}}\]