## Calculus Of One Real Variable – By Pheng Kim Ving Chapter 13: Plane Curves – Section 13.2.4: Arc Length And Area Of Surface Of Revolution Of Polar Curves

13.2.4
Arc Length And Area Of Surface Of Revolution Of Polar Curves

 1. Arc Length Of Polar Curves

 Fig. 1.1   Arc Length Of A Polar Curve.

Using Parametrization

In Section 13.1.3 we see that the arc length s of the parametric curve x = f(t), y = g(t) from t = a to t = b is:

Using Differential Triangle

Refer to Fig. 1.2. In Section 12.6 Part 3 we see that, for the arc length s of a function with Cartesian equation y = f(x), the
differential arc length at any point x is ds, which is the line segment of the tangent at x and is the hypotenuse of the
differential triangle whose legs are dx and dy at x, so that:

# Differential of arc length is:

 Fig. 1.3   Differential Triangle PAB:

Arc Length

Example 1.1

 Fig. 1.4   Total Length Of A Cardioid.

Solution
By symmetry the required length s is twice the length of the top-half cardioid and so is:

EOS

Had we not utilized symmetry then we would have:

which is more complicated.

 2. Area Of Surface Of Revolution Of Polar Curves

{2.1} Section 12.7.

# Revolution About y-Axis.

For the revolution about the y-axis, as displayed in Fig. 2.2, the area S of the surface is:

Example 2.1

 Fig. 2.3   Calculating Area Generated By Revolving A Cardioid About x-Axis.

Solution
By symmetry the given surface is the same as the surface generated by revolving the top-half cardioid about the x-axis.
So the desired area S is:

EOS

which is incorrectly double the correct answer. This is because there are two coinciding surfaces: one generated by the
top-half cardioid and the other by the bottom-half.

# Problems & Solutions

Solution

By symmetry the required length s is twice the length of the left-half of the cardioid and so is:

Solution

The desired length s is:

Note

Here the problem statement must specify the range of the angle, because the curve doesn’t repeat itself. In
Problem & Solution 1 the problem statement doesn’t have to do so if it considers the entire curve, because the curve repeats
itself.

Solution

The wanted length s is:

Note

Solution

The required area S is:

Solution

By symmetry the given surface is the same as the surface obtained by rotating the left-half of the cardioid around the
y-axis. So the desired area S is: