Problem 72
For the following equations, determine which of the conic sections is described. $$ x^{2}+2 \sqrt{3} x y+3 y^{2}-6=0 $$
Problem 72
For the following exercises, find the area of the surface obtained by rotating the given curve about the \(x\) -axis\(x=a \cos ^{3} \theta, \quad y=a \sin ^{3} \theta, \quad 0 \leq \theta \leq \frac{\pi}{2}\)
Problem 73
For the following equations, determine which of the conic sections is described. $$ x^{2}-x y+y^{2}-2=0 $$
Problem 73
There is a curve known as the "Black Hole. " Use technology to plot \(r=e^{-0.01 \theta}\) for \(-100 \leq \theta \leq 100\).
Problem 73
[T] Use a CAS to find the area of the surface generated by rotating \(x=t+t^{3}, y=t-\frac{1}{t^{2}}, 1 \leq t \leq 2\) about the \(x\) -axis. (Answer to three decimal places.)
Problem 73
Find the points at which the following polar curves have a horizontal or vertical tangent line.The cardioid \(r=1+\sin \theta\)
Problem 74
Find the points at which the following polar curves have a horizontal or vertical tangent line.Show that the curve \(r=\sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x=1\) as a vertical asymptote.
Problem 74
For the following equations, determine which of the conic sections is described. $$ 34 x^{2}-24 x y+41 y^{2}-25=0 $$
Problem 74
Find the surface area obtained by rotating \(x=3 t^{2}, y=2 t^{3}, 0 \leq t \leq 5\) about the \(y\) -axis.
Problem 75
For the following equations, determine which of the conic sections is described. $$ 52 x^{2}-72 x y+73 y^{2}+40 x+30 y-75=0 $$
