91Ó°ÊÓ

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

Give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation in Cartesian coordinates. Eccentricity: Vertices: (0,±1)

Sketch the region defined by the inequalities \(-1 \leq r \leq 2\) and \(-\pi / 2 \leq \bar{\theta} \leq \pi / 2\).

Find a parametrization for the curve. the ray (half line) with initial point (2,3) that passes through the point (-1,-1)

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$-\pi / 2 \leq \theta \leq \pi / 2, \quad 1 \leq r \leq 2$$

Find a parametrization for the curve. The ray (half line) with initial point (-1,2) that passes through the point (0,0)

Graph the sets of points whose polar coordinates satisfy the equations and inequalities. $$0 \leq \theta \leq \pi / 2, \quad 1 \leq|r| \leq 2$$

Find the lengths of the curves. The parabolic segment \(r=2 /(1-\cos \theta), \quad \pi / 2 \leq \theta \leq \pi\)

Give the eccentricities and the vertices or foci of hyperbolas centered at the origin of the \(x y\) -plane. In each case, find the hyperbola's standard-form equation in Cartesian coordinates. Eccentricity: 2 Vertices: (±2,0)

Sketch the region defined by the inequalities \(0 \leq r \leq 2 \sec \theta\) and \(-\pi / 4 \leq \theta \leq \pi / 4\).