Calculating Area: Surface of Revolution
Calculating the area of a surface of revolution is a significant topic in calculus, focusing on the surface generated when a curve is rotated around an axis. This concept is foundational in mathematics and has practical applications in fields like physics and engineering. To determine the surface area, one typically utilizes integration techniques based on the lateral surface area of a frustum of a cone. The process begins by considering a continuous function defined over a specific interval and partitioning that interval into subintervals. Each segment, when revolved about the axis, forms a frustum, allowing for the calculation of the lateral surface area.
The overall surface area is found by integrating the differential surface area across the defined interval. While the x-axis is a common axis of rotation, surfaces can also be generated by rotating curves around other lines. The procedure can involve complex integration techniques such as trigonometric substitution, particularly when dealing with more intricate functions. Understanding these concepts not only enriches mathematical knowledge but also enhances problem-solving skills in various scientific disciplines.
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Calculating Area: Surface of Revolution
Determining surface area is an important application of calculus. In Euclidean space, a surface of a revolution is the surface generated by a curve rotated around an axis (straight line) in its plane.
Before discussing the main topic, it is useful to briefly discuss a related topic: arc length. Figure 1 below shows the graph of
, on which two points
and
have been labeled. Suppose that these points are infinitesimally close together.
The curve is obviously not linear, but if one zooms in for a closer look the curve is approximately linear between
and
. See Figure 2.
The (approximately) linear part of the curve between
and
is a differential unit of arc length of the curve, and its length has been labeled
. This segment can be thought of as the hypotenuse of a right triangle, whose legs of length
and
are labeled. By the Pythagorean theorem we have
, or
A surface of revolution is a surface in space generated by revolving a curve in the plane about an axis. For example revolving the semi-ellipse whose equation is 
(see Figure 3 ) about the x-axis generates an ellipsoid (see Figure 3 and Figure 4).
Calculating the area of a surface of revolution, such as that in Figure 4, is one of the many applications of integration that is used not only in mathematics, but also in the physical and engineering sciences.
Overview
The method of calculating surface area of revolution used here is based on the formula for the lateral surface area of the frustum of a cone. See Figure 5.
The lateral surface area (that is, the area around the slanted body) is given by
Let
be a continuous function on an interval
, and let
any partition of
.
Let the widths of the subintervals defined by
be
, respectively. See Figure 6.
The points
have been marked, as have the chords joining these points. Their lengths have been labeled
. When the
chord is revolved about the x-axis, the surface that it generates is the frustum of a cone with
,
, and
. The lateral surface area
of the
frustum can be found using
:
Expressing
in terms of differentials and neglecting second order differentials, the differential surface area element is
. Using
to eliminate
, the final result for
is obtained:
The total area
of the surface generated by revolving the graph of
on
about the x-axis is found by integrating
from
to
:
A surface of revolution can be obtained by revolving the graph of a continuous function about any line, not just the x-axis.
As an example, equation
can be used to compute the area of the ellipsoid from earlier in the article. The curve given by the graph of
on the interval
is revolved about the x-axis to generate the surface. Noting that
the following is obtained.
Many steps were omitted in obtaining the final result from
. The integration is rather involved and requires a technique known as trigonometric substitution.
Bibliography
Anton, Howard, Irl Bivens, and Stephen Davis. Calculus. Hoboken: Wiley, 2012. Print.
Briggs, William L. Calculus for Scientists and Engineers: Early Transcendentals. Boston: Pearson, 2013. Print.
Edwards, C. H. Calculus: Early Transcendentals. Boston: Pearson, 2014. Print.
Larson, Ron, and Bruce H. Edwards. Calculus. Boston: Cengage, 2014. Print.
Stewart, James. Calculus: Early Transcendentals. Belmont, CA: Cengage, 2012. Print.