91Ó°ÊÓ

Find a parametrization for the curve. the lower half of the parabola \(x-1=y^{2}\)

Which of the following has the same graph as \(r=1-\cos \theta ?\) a. \(r=-1-\cos \theta\) b. \(r=1+\cos \theta\) Confirm your answer with algebra.

Find the areas of the surfaces generated by revolving the curves about the indicated axes. $$x=\ln (\sec t+\tan t)-\sin t, y=\cos t, 0 \leq t \leq \pi / 3 ; x-\operatorname{axis}$$

Which of the following has the same graph as \(r=\cos 2 \theta ?\) a. \(r=-\sin (2 \theta+\pi / 2)\) b. \(r=-\cos (\theta / 2)\) Confirm your answer with algebra.

Find a parametrization for the curve. the left half of the parabola \(y=x^{2}+2 x\)

Graph the equation \(r=1-2 \sin 3 \theta\)

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r^{2}=1$$

Find a parametrization for the curve. the ray (half line) with initial point ( 2,3 ) that passes through the point (-1,-1)

A cone frustum The line segment joining the points (0,1) and (2, 2) is revolved about the \(x\) -axis to generate a frustum of a cone. Find the surface area of the frustum using the parametrization \(x=2 t, y=t+1,0 \leq t \leq 1 .\) Check your result with the geometry formula: Area \(=\pi\left(r_{1}+r_{2}\right)(\text { slant height })\)

A cone The line segment joining the origin to the point \((h, r)\) is revolved about the \(x\) -axis to generate a cone of height \(h\) and base radius \(r .\) Find the cone's surface area with the parametric equations \(x=h t, y=r t, 0 \leq t \leq 1 .\) Check your result with the geometry formula: Area \(=\pi r(\text { slant height })\)