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

 

 

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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:

 

 

Fig. 1.2

 

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.

 

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2. Area Of Surface Of Revolution Of Polar Curves

 

 

{2.1} Section 12.7.

 

Fig. 2.1

 

Revolution About x-Axis.

 

Fig. 2.2

 

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.

 

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Problems & Solutions


 

 

Solution

 

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

 

 

 

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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.

 

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Solution

 

The wanted length s is:

 

 

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Note

 

 

Solution

 

The required area S is:

 

 

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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:

 

 

 

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