Essential Concepts
- The area of a region in polar coordinates defined by the equation [latex]r=f\left(\theta \right)[/latex] with [latex]\alpha \le \theta \le \beta[/latex] is given by the integral [latex]A=\frac{1}{2}{{\displaystyle\int }_{\alpha }^{\beta }\left[f\left(\theta \right)\right]}^{2}d\theta[/latex].
- To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
- The arc length of a polar curve defined by the equation [latex]r=f\left(\theta \right)[/latex] with [latex]\alpha \le \theta \le \beta[/latex] is given by the integral [latex]L={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime }\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}}d\theta[/latex].
Key Equations
- Area of a region bounded by a polar curve
[latex]A=\frac{1}{2}{\displaystyle\int }_{\alpha }^{\beta }{\left[f\left(\theta \right)\right]}^{2}d\theta =\frac{1}{2}{\displaystyle\int }_{\alpha }^{\beta }{r}^{2}d\theta[/latex] - Arc length of a polar curve
[latex]L={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime }\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}}d\theta[/latex]
Candela Citations
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- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction