91Ó°ÊÓ

Find the lengths of the curves. $$\text { The spiral } r=e^{\theta} / \sqrt{2}, \quad 0 \leq \theta \leq \pi$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26\). $$0 \leq \theta \leq \pi, \quad r=-1$$

Find the lengths of the curves. $$\text { The cardioid } r=1+\cos \theta$$

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26\). $$\pi / 4 \leq \theta \leq 3 \pi / 4, \quad 0 \leq r \leq 1$$

Find the area enclosed by the ellipse $$x=a \cos t, \quad y=b \sin t, \quad 0 \leq t \leq 2 \pi$$

Find the area under \(y=x^{3}\) over [0,1] using the following parametrizations. a. \(x=t^{2}, y=t^{6}\) b. \(x=t^{3}, \quad y=t^{9}\)

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26\). $$-\pi / 4 \leq \theta \leq \pi / 4, \quad-1 \leq r \leq 1$$

Find the lengths of the curves. $$\text { The curve } r=a \sin ^{2}(\theta / 2), \quad 0 \leq \theta \leq \pi, \quad a>0$$

Graph the limaçons in Exercises. Limaçon ("lee-ma-sahn") is Old French for "snail." You will understand the name when you graph the limaçons in Exercise \(25 .\) Equations for limaçons have the form \(r=a \pm b \cos \theta\) or \(r=a \pm b \sin \theta .\) There are four basic shapes. Limaçons with an inner loop a. \(r=\frac{1}{2}+\cos \theta\) b. \(r=\frac{1}{2}+\sin \theta\)

Find the lengths of the curves $$x=\cos t, \quad y=t+\sin t, \quad 0 \leq t \leq \pi$$