91Ó°ÊÓ

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=\frac{5}{\sin \theta-2 \cos \theta}$$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r^{2} \sin 2 \theta=2$$

A wheel of radius \(a\) rolls along a horizontal straight line without slipping. Find parametric equations for the curve traced out by a point \(P\) on a spoke of the wheel \(b\) units from its center. As parameter, use the angle \(\theta\) through which the wheel turns. The curve is called a trochoid, which is a cycloid when \(b=a\)

Find the coordinates of the centroid of the curve $$ x=\cos t, \quad y=t+\sin t, \quad 0 \leq t \leq \pi $$

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=\cot \theta \csc \theta$$

Find the point on the parabola \(x=t, y=t^{2},-\infty

Find the point on the ellipse \(x=2 \cos t, y=\sin t, 0 \leq t \leq 2 \pi\) closest to the point \((3 / 4,0) .\) (Hint: Minimize the square of the distance as a function of \(t .\) )

Replace the polar equations with equivalent Cartesian equations. Then describe or identify the graph. $$r=4 \tan \theta \sec \theta$$

Most centroid calculations for curves are done with a calculator or computer that has an integral evaluation program. As a case in point, find, to the nearest hundredith, the coordinates of the centroid of the curve $$ x=t^{3}, \quad y=3 t^{2} / 2, \quad 0 \leq t \leq \sqrt{3} $$

Most centroid calculations for curves are done with a calculator or computer that has an integral evaluation program. As a case in point, find, to the nearest hundredth, the coordinates of the centroid of the curve $$ x=t^{3}, \quad y=3 t^{2} / 2, \quad 0 \leq t \leq \sqrt{3} $$