Surface Area of Solid of Revolution

(You need Java Runtime Environment (JRE) to run GraphFunc applet in this website)

Show some examples of finding the surface areas of solids of revolution defined in Cartesian, Polar and Parametric coordinates.  Then use the GraphFunc utility online to confirm its results.  See the demo.

Surface Area with Cartesian Coordinates

Definition  If the portion of the curve y = f(x) between x = a and x = b is revolved about the x-axis, the area S of the surface generated is given by the following:

Example 1. Find the area of the surface obtained by rotating the curve y = x, , about the x-axis.

Solution.

This is a straightforward computation using the formula for the surface area.  We have

( =  71.08612701053…)

 Use the GraphFunc utility online to confirm the above result as shown in the following steps:

Begin by going to http://graph.seriesmathstudy.com, and wait for the GraphFunc applet to be loaded into your computer.  Then do the following steps:

1. Select the View 3D item from  drop-down list.
2. Enter the expression x at the command line.
3. Press on the Graph It! button to plot the graph.
4. Select the x-axis item fromdrop-down list box.  It shows the graph be rotated around the x-axis.
5. Enter the limit values x = 0 and x = 4 at the label.
6. Press on the Surface Area button to compute the area of surface solid of revolution.  The result is shown in Figure 1.

Figure 1.

Example 2. Find the area of the surface obtained by rotating the curve, , about the x-axis.

Solution.

Since, we substitute y’ into (I) to get

Use , we get

( = 7.9876494805994…)

 Use GraphFunc utility to verify the above result by following the steps that are indicated in the example 1.  The surface area of the given curve is computed and depicted in the Figure 2 below.

Figure 2.

## Surface Area with Polar Coordinates

Definition The surface area of the region obtained by rotatingfromto about the polar axis is given by

If the curve is revolved about the y-axis, the formula of surface area is

Example 3. Find the area of the surface formed by revolving the curvefromtoabout the polar axis.

Solution.

We use the formula (II) with, we have

(= 12.5663706143591...)

 The steps below show using GraphFunc utility online to confirm the above result as described in Figure 3.

Begin by going to http://graph.seriesmathstudy.com, and wait for the applet to be loaded into your computer.  Then

1. Select the View 3D item from  drop-down list box.

1. Select the Polar item fromdrop-down list box. (An important step.)

1. Enter the expression 2*sin(t) at the command line.

1. Press on the Graph It! button to plot the graph. (Note:.)

1. Select the x-axis item fromdrop-down list box.  It shows the graph be rotated around the polar axis (or x-axis).

1. Enter the values x = 0 and x = pi/2 at the label .

1. Press on the Surface Area button to compute the area of surface solid of revolution.  Click on the Zoom In button.  The Figure 3 shows the result.

Figure 3.

## Area of Surface of Revolution in Parametric Form

Definition If a smooth curve C given by x = f(t) and y = g(t) does not cross itself on an interval , then the area S of the surface of revolution formed by revolving C about the polar axis is given by

Example 4.  Find the area of the surface generated by revolving the curve x = t, y = 2t, from t = 0 to t = 4 about the polar axis (or x-axis).

### Solution

We use the formula (III) with g(t) = 2t, and , we have

.

 The Figure 4 below shows using the GraphFunc utility online to compute the surface area and gives the approached result.  Note that user must select the Parametric item from Cartesian drop-down list box first in order to switch the mode from either Cartesian or Polar coordinates to Parametric coordinates.

Figure 4. The surface area of given graph is depicted by the curves in pink from t = 0 to t = 4.