In polar coordinates, we may wish to integrate a function [math]f\left(r,\theta\right)[/math] over a region bounded on the "inside" by the curve [math]r=g_1\left(\theta\right)[/math], bounded on the "outside" by the curve [math]r=g_2\left(\theta\right)[/math], and where [math]\theta[/math] ranges from [math]\theta=\alpha[/math] to [math]\theta=\beta[/math]. In this case we find that [br] [math]\int\int_Rf\left(r,\theta\right)dA=\int_{\alpha}^{\beta}\left[\int_{g_1\left(\theta\right)}^{g_2\left(\theta\right)}f\left(r,\theta\right)r\text{ }dr\right]d\theta[/math].[br][br]Don't forget the extra [math]r[/math] that appears in the inside integral! In the first example that loads with the interactive figure, [math]g_1\left(\theta\right)=1[/math], [math]g_2\left(\theta\right)=1+\cos\left(\theta\right)[/math], [math]\alpha=-\frac{\pi}{3}[/math], and [math]\beta=\frac{\pi}{6}[/math].[br][br][b]Note on entering Greek symbols[/b]. You can easily enter [math]\theta[/math] and [math]\pi[/math] in the text fields above by typing Alt+t and Alt+p on your keyboard. Also observe that as you type an equation a symbol pop-up menu is available.[br][br]

[i]Developed for use with Thomas' Calculus, published by Pearson.[/i]