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 by type graphs (Cartesian, Polar and Parametric coordinates). Then use the GraphFunc utility online to confirm the 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



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( = 71.08612701053…)
|
Use the GraphFunc utility online to confirm the above result as shown in the following steps: |
Get into the website at http://graph.seriesmathstudy.com, and wait for the GraphFunc applet to be automatically loaded online. Then
drop-down list.
drop-down
list box. It shows the graph be
rotated around the x-axis.

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…)
|
Follow the steps shown in Example 1. The surface area that GraphFunc utility computes is depicted in Figure 2. |

Figure 2.
Definition The
surface area of the region obtained by rotating
from
to
about the polar axis is given by

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

Example 3. The area
of the surface formed by revolving the curve
from
to
about
the polar axis.
Solution.
We use the formula (II) with
, we
have


![]()
![]()
(= 12.5663706143591...)
|
Use the GraphFunc utility online to confirm the above result as shown in the following steps:: |
Get into the website at http://graph.seriesmathstudy.com and wait for the applet to be automatically loaded online. Then
drop-down list box.
drop-down
list box. (An important step.)
drop-down
list box. It shows the graph be
rotated around the polar axis (or x-axis).

Figure 3.
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).
We use the formula (III) with g(t) = 2t,
and
, we
have
.
|
Use the GraphFunc utility online to compute the surface area and its computation is shown in the Figure 4. Note that user must select the Parametric item from Cartesian drop-down list box to switch the mode from either Cartesian or Polar coordinates to Parametric coordinates. |

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