exercise:Ba0836a760

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The curve defined by the equation [math]r=\frac1{1+\cos\theta}[/math] in polar coordinates is a parabola similar to the one discussed in Example.

• Draw the parabola, and show the region [math]R[/math] bounded by this curve and the line [math]\theta=\frac\pi2[/math].
• Express [math]\mbox{''area''}(R)[/math] as a definite integral using the integral formula for area in polar coordinates.
• Evaluate the integral in part (b) using the trigonometric substitution [math]z=\tan \frac\theta2[/math] (see equation) and the Change of Variable Theorem for Definite Integrals.
• Write this curve as the graph of an equation in [math]x[/math]- and [math]y[/math]-coordinates, and thence compute area(R).