91Ó°ÊÓ

Find the a:ea of the surface generated by revolving the curve about the \(x\) -axis. \(f(x)=\sqrt{x} . \quad x \in[1,2]\).

Express the curve by an equation in \(x\) and \(y\). $$x(t)=\csc t, \quad y(t)=\cot t$$

Find an equation in \(x\) and \(y\) for the line tangent to the polar curve at the indicated value of \(\theta\). $$r=\frac{5}{4-\cos \theta} \quad \theta=\frac{1}{6} \pi$$

Give the rectangular coordinates of the point. $$\left[1, \frac{1}{4} \pi \right]$$

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse. $$3 x^{2}+4 y^{2}-12=0$$

Find the length of the graph and compare it to the straight-line distance between the endpoints of the graph. $$f(x)=\frac{1}{10} x^{5}+\frac{1}{6} x^{-3}, \quad x \in[1,2]$$

An ellipse is given. Find the center, the foci, the length of the major axis, and the length of the minor axis. Then sketch the ellipse. $$4 x^{2}+9 y^{2}-18 y=27$$

Sketch the polar curve. $$r^{2}=\sin 2 \theta.$$

Find the length of the graph and compare it to the straight-line distance between the endpoints of the graph. $$f(x)=\ln (\sec x), \quad x \in\left[0, \frac{1}{4} \pi\right]$$

Find the area between the curves. $$\begin{aligned} &-r=e^{\theta}, \quad 0 \leq \theta \leq \pi ; \quad r=\theta, \quad 0 \leq \theta \leq \pi ; \quad \text { the rays }\\\ &\theta=0, \quad \theta=\pi \end{aligned}$$