Question

(a)[PR1] How do you find the slope of a tangent line to a polar curve?

(b) How do you find the area of a region bounded by a polar curve?

(c) How do you find the length of a polar curve?

Short answer

1. For a polar curve the slope of the tangent is\({\rm{M = }}\frac{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}} \cdot {\rm{sin(\theta ) + r}} \cdot {\rm{cos(\theta )}}}}{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}} \cdot {\rm{cos(\theta ) - r}} \cdot {\rm{sin(\theta )}}}}\).
2. For a polar curve the of a region is\({\rm{A = }}\int_{\rm{\alpha }}^{\rm{\beta }} {\frac{{\rm{1}}}{{\rm{2}}} \cdot } {{\rm{r}}^{\rm{2}}}{\rm{d\theta }}\).
3. For a polar curve the expression for its length is \({\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}} \right)}^{\rm{2}}}} {\rm{d\theta }}} \).

Step-by-step solution

Concept Introduction

Polar curves are graphs of equations with polar coordinates as their basis. The polar curve, like ordinary equations and curves, is made up of all polar coordinates that satisfy the specified equation.

Slope of tangent to Polar Curve

(a)

Mathematically polar curve is defined as –

\(\begin{aligned}{c}{\rm{f(x,y) = f(r,\theta )}}\\{\rm{ = rcos(\theta ) + rsin(\theta )}}\end{aligned}\)

Now tangent is basically –

\(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}{\rm{ = }}\frac{{{{{\rm{dy}}} \mathord{\left/

{\vphantom {{{\rm{dy}}} {{\rm{d\theta }}}}} \right.

\kern-\nulldelimiterspace} {{\rm{d\theta }}}}}}{{{{{\rm{dx}}} \mathord{\left/

{\vphantom {{{\rm{dx}}} {{\rm{d\theta }}}}} \right.

\kern-\nulldelimiterspace} {{\rm{d\theta }}}}}}\)

Hence, the slope of tangent\({\rm{M}}\)will be –

\({\rm{M = }}\frac{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}} \cdot {\rm{sin(\theta ) + r}} \cdot {\rm{cos(\theta )}}}}{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}} \cdot {\rm{cos(\theta ) - r}} \cdot {\rm{sin(\theta )}}}}\)

Therefore, the result is obtained as \({\rm{M = }}\frac{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}} \cdot {\rm{sin(\theta ) + r}} \cdot {\rm{cos(\theta )}}}}{{\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}} \cdot {\rm{cos(\theta ) - r}} \cdot {\rm{sin(\theta )}}}}\).

Area of region bounded by Polar Curve

(b)

Area of polar coordinates (say\({\rm{A}}\)) is calculated by standard formula –

\({\rm{A = }}\int_{\rm{\alpha }}^{\rm{\beta }} {\frac{{\rm{1}}}{{\rm{2}}} \cdot } {{\rm{r}}^{\rm{2}}}{\rm{d\theta }}\)

This gives the area enclosed by below figure –

Therefore, the result is obtained as \({\rm{A = }}\int_{\rm{\alpha }}^{\rm{\beta }} {\frac{{\rm{1}}}{{\rm{2}}} \cdot } {{\rm{r}}^{\rm{2}}}{\rm{d\theta }}\).

Length of a Polar Curve

(c)

Mathematically polar curve is defined as –

\({\rm{r = f(\theta ), \theta }} \in \left( {{\rm{a,b}}} \right)\)

Then the length of polar curve (say \({\rm{L}}\)) is –

\({\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}} \right)}^{\rm{2}}}} {\rm{d\theta }}} \)

Therefore, the result is obtained as \({\rm{L = }}\int_{\rm{a}}^{\rm{b}} {\sqrt {{{\rm{r}}^{\rm{2}}}{\rm{ + }}{{\left( {\frac{{{\rm{dr}}}}{{{\rm{d\theta }}}}} \right)}^{\rm{2}}}} {\rm{d\theta }}} \).