91Ó°ÊÓ

Find the length of the graph and compare it to the straight-line distance between the endpoints of the graph. $$f(x)=\frac{1}{2} x \sqrt{3-x^{2}}+\frac{3}{2} \arcsin \left(\frac{1}{3} \sqrt{3} x\right), \quad x \in[0,1]$$

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola. $$x^{2}-y^{2}=1$$

Rcpresent the area by one or more integrals. Outside \(r=2,\) but inside \(r=4 \sin \theta\)

Find the a:ea of the surface generated by revolving the curve about the \(x\) -axis. \(r=e^{\theta}, \quad \theta \in\left[0, \frac{1}{2} \pi\right]\).

Below some points are specified in rectangular coordinates. Give all possible polar coordinates for each point. $$(0,1)$$

Below some points are specified in rectangular coordinates. Give all possible polar coordinates for each point. $$(1,0)$$

Parametrize the curve by a pair of differentiable functions $$x=x(t), \quad y=y(t) \quad \text { with } \quad\left[x^{\prime}(t)\right]^{2}+\left[y^{\prime}(t)\right]^{2} \neq 0$$ Sketch the curve and determine the tangent line at the origin from the parametrization thal you selected. $$y^{3}=x^{5}$$

A hyperbola is given. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis. Then sketch the hyperbola. $$y^{2}-x^{2}=1$$

Find the length of the graph and compare it to the straight-line distance between the endpoints of the graph. $$f(x)=\ln (\sin x), \quad x \in\left[\frac{1}{6} \pi, \frac{1}{2} \pi\right]$$

Find the a:ea of the surface generated by revolving the curve about the \(x\) -axis. \(y=\cosh x, \quad x \in[0, \ln 2]\).