91Ó°ÊÓ

Find the surface area of the torus generated by revolving the circle given by \(r=2\) about the line \(r=5 \sec \theta\)

The curve represented by the equation \(r=a \theta,\) where \(a\) is a constant, is called the spiral of Archimedes. (a) Use a graphing utility to graph \(r=\theta,\) where \(\theta \geq 0\). What happens to the graph of \(r=a \theta\) as \(a\) increases? What happens if \(\theta \leq 0 ?\) (b) Determine the points on the spiral \(r=a \theta(a>0, \theta \geq 0)\) where the curve crosses the polar axis. (c) Find the length of \(r=\theta\) over the interval \(0 \leq \theta \leq 2 \pi\). (d) Find the area under the curve \(r=\theta\) for \(0 \leq \theta \leq 2 \pi\).

Find the centroid of the region bounded by the graph of the parametric equations and the coordinate axes. $$ x=\sqrt{t}, y=4-t $$