Question

How will you find the length of the cardioid \[r = 1 - \cos \theta \] ?

Answer 1

Hint:Here as we need to find the length of the cardioid so we will use the formula of the length \[(L)\] of the cardioids by \[L = 2\int\limits_0^\pi {\sqrt {\left( {{r^2} + {{\left( {\dfrac{{dr}}{{d\theta }}} \right)}^2}} \right)} } d\theta \] . We will solve this expression in order to calculate the length of the cardioids by substitution of the value of \[r\] and \[\dfrac{{dr}}{{d\theta }}\] then find the correct limits of integration also.

Formula used:
The formula to find the length of the cardioid is \[L = 2\int\limits_0^\pi {\sqrt {\left( {{r^2} + {{\left( {\dfrac{{dr}}{{d\theta }}} \right)}^2}} \right)} } d\theta \]
where \[L\] represents the length of the cardioid.

Complete step by step answer:
As we know that in above question asked we need to find the length of the cardioid \[r = 1 - \cos \theta \]
\[\dfrac{{dr}}{{d\theta }} = \sin \theta \]
By simplifying we will get
\[ds = \sqrt {{{\sin }^2}\theta + {{(1 - \cos \theta )}^2}d\theta } \\
\Rightarrow \sqrt {{{\sin }^2}\theta + 1 - 2\cos \theta + {{\cos }^2}\theta d\theta } \\
\Rightarrow \sqrt 2 \sqrt {\left( {1 - \left( {1 - 2{{\sin }^2}\dfrac{\theta }{2}} \right)} \right)} d\theta \\
\Rightarrow \sqrt 2 \sqrt {2{{\sin }^2}\left( {\dfrac{\theta }{2}} \right)} d\theta \\
\Rightarrow 2\sin \left( {\dfrac{\theta }{2}} \right)d\theta \\ \]
So if we assume that length is of one full revolution
\[L = \int_0^{2\pi } {\sin \left( {\dfrac{\theta }{2}} \right)} d\theta \\
\Rightarrow 2\left[ { - 2\cos \dfrac{\theta }{2}} \right]_0^{2\pi } \\
\Rightarrow 4\left[ {\cos \dfrac{\theta }{2}} \right]_{2\pi }^0 \\
\Rightarrow 4\left[ {1 - ( - 1)} \right] \\
\therefore L = 8 \\ \]
Hence, the length of the cardioid is \[8\].

Additional information:
A cardioid is a plane figure having a heart shaped curve and is symmetrical about the initial line. We can represent the equation of the cardioid in polar form and later we can convert it into a Cartesian coordinate system also.

Note:While solving such types of questions easily we need to have some understanding about trigonometric properties. Some of the trigonometric properties that is basic \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] and \[1 + \cos \theta = 2co{s^2}\dfrac{\theta }{2}\]. Remember that the shape of the cardioid is formed by tracing a point on the boundary of a circle and then rolling onto another circle of the same radius. In the above equation given, keep in mind that \[\theta \] represents the polar angle and while solving we need to convert polar form into Cartesian structure.