91Ó°ÊÓ

Find the area of the surface generated by revolving the curve \(x=2+\cos t, y=1+\sin t\), for \(0 \leq t \leq 2 \pi\) about the \(x\)-axis.

If two tangent lines to the ellipse \(9 x^{2}+4 y^{2}=36\) intersect the \(y\)-axis at \((0,6)\), find the points of tangency.

Investigate the family of curves defined by the polar equations \(r=|\cos n \theta|\), where \(n\) is some positive integer. How do the number of leaves depend on \(n\) ?

If the tangent lines to the hyperbola \(9 x^{2}-y^{2}=36\) intersect the \(y\)-axis at \((0,6)\), find the points of tangency.

Find the area of the surface generated by revolving the curve \(x=(2 / 3) t^{3 / 2}, y=2 \sqrt{t}\), for \(0 \leq t \leq 2 \sqrt{3}\) about the \(y\)-axis.

Polar graphs can be used to represent different spirals. The spirals can unwind clockwise or counterclockwise. Find the condition on \(c\) to make the spiral of Archimedes, \(r=c \theta\), unwind clockwise and counterclockwise.

The slope of the tangent line to the hyperbola $$ 2 x^{2}-7 y^{2}-35=0 $$ at two points on the hyperbola is \(-\frac{2}{3}\). What are the coordinates of the points of tangency?

Find the area of the surface generated by revolving the curve \(x=t+\sqrt{7}, y=t^{2} / 2+\sqrt{7} t\), for \(-\sqrt{7} \leq t \leq \sqrt{7}\) about the \(y\)-axis.

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0)\). As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Sketch the reciprocal spiral given by \(r=c / \theta\). For \(c>0\), does it unwind in the clockwise direction?