Return To Contents
Go To Problems & Solutions
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 tophalf cardioid and so is:
EOS
Had we not utilized symmetry then we would have:
which is more complicated.
Return To Top Of Page Go To Problems & Solutions
2. Area Of Surface Of Revolution Of Polar Curves 
^{{}^{2.1}} Section 12.7.

Fig.
2.1
Revolution About xAxis. 

Fig.
2.2
Revolution
About yAxis.

For the revolution about the yaxis, 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 xAxis. 
Solution
By symmetry the given surface is the same as the surface generated by revolving
the tophalf cardioid about the xaxis.
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
tophalf cardioid and the other by the bottomhalf.
Problems & Solutions 
Solution
By symmetry the required length s is twice the length of the lefthalf 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 lefthalf of the cardioid around the
yaxis. So the desired area S is:
Return To Top Of Page Return To Contents