Question

How do you find the exact length of the polar curve \[r = {e^\theta }\]?

Answer 1

Hint: We use the concept that the function given is a curve. Use the formula of arc length of a curve. Calculate the differentiation of the curve with respect to ‘r’ and substitute the values in the formula of arc length.
* For a curve \[r = f(\theta )\], where \[{\theta _1} < \theta < {\theta _2}\], the arc length is given by the formula:
\[L = \int\limits_{{\theta _1}}^{{\theta _2}} {\sqrt {{r^2} + {{\left( {\dfrac{{dr}}{{d\theta }}} \right)}^2}} } d\theta \]
* General formula of differentiation is \[\dfrac{d}{{dx}}({x^n}) = n{x^{n - 1}}\]

Complete step by step solution:
Here we have function \[r = {e^\theta }\]
First we calculate the differentiation of the given function with respect to ‘r’.
\[ \Rightarrow \dfrac{{dr}}{{d\theta }} = \dfrac{d}{{d\theta }}\left( {{e^\theta }} \right)\]
\[ \Rightarrow \dfrac{{dr}}{{d\theta }} = {e^\theta }\]
Now we know the arc length L is given by the formula \[L = \int\limits_{{\theta _1}}^{{\theta _2}} {\sqrt {{r^2} + {{\left( {\dfrac{{dr}}{{d\theta }}} \right)}^2}} } d\theta \]
Substitute the value of \[r = {e^\theta }\]and \[\dfrac{{dr}}{{d\theta }} = {e^\theta }\] in the formula.
\[ \Rightarrow L = \int\limits_{{\theta _1}}^{{\theta _2}} {\sqrt {{{\left( {{e^\theta }} \right)}^2} + {{\left( {{e^\theta }} \right)}^2}} } d\theta \]
Square the terms inside the brackets in right hand side of the equation.
\[ \Rightarrow L = \int\limits_{{\theta _1}}^{{\theta _2}} {\sqrt {{e^{2\theta }} + {e^{2\theta }}} } d\theta \]
Add the terms under the square root in right hand side of the equation.
\[ \Rightarrow L = \int\limits_{{\theta _1}}^{{\theta _2}} {\sqrt {2{e^{2\theta }}} } d\theta \]
Now we cancel the square root from the square power wherever possible in right hand side of the equation.
\[ \Rightarrow L = \int\limits_{{\theta _1}}^{{\theta _2}} {{e^\theta }\sqrt 2 } d\theta \]
Bring the constant value out of the integration in right hand side of the equation
\[ \Rightarrow L = \sqrt 2 \int\limits_{{\theta _1}}^{{\theta _2}} {{e^\theta }} d\theta \]
Integrate the function in right hand side of the equation
\[ \Rightarrow L = \sqrt 2 \left[ {{e^\theta }} \right]_{{\theta _1}}^{{\theta _2}}\]
Apply the limits of the angle in right hand side of the equation
\[ \Rightarrow L = \sqrt 2 \left[ {{e^{{\theta _2}}} - {e^{{\theta _1}}}} \right]\]
\[\therefore \]The exact length of the polar curve \[r = {e^\theta }\]is \[\sqrt 2 \left[ {{e^{{\theta _2}}} - {e^{{\theta _1}}}} \right]\]where \[{\theta _1} < \theta < {\theta _2}\].

Note: Many students make the mistake of differentiating the curve with respect to ‘r’, which is wrong. As we can see the function r is dependent on the angle, so we differentiate the function or curve with respect to the angle. Also, if the range of angle is given in the question, students are advised to apply the limits in the end and calculate the answer using a calculator.