91Ó°ÊÓ

Arc Length In Exercises \(45-50\) , find the are length of the curve on the given interval. $$ \text{Parametric Equations} \quad \text{Interval} $$ $$ x=6 t^{2}, \quad y=2 t^{3} \quad 1 \leq t \leq 4 $$

In Exercises 41–48, use the results of Exercises 37–40 to find a set of parametric equations for the line or conic. Ellipse: vertices: \(( \pm 10,0) ;\) foci: \(( \pm 8,0)\)

Finding an Equation of a Hyperbola In Exercises \(41-48,\) find an equation of the hyperbola. Center: \((0,0)\) Vertex: \((6,0)\) Focus: \((10,0)\)

The radiation from a transmitting antenna is not uniform in all directions. The intensity from a particular antenna is modeled by \(r=a \cos ^{2} \theta\) (a) Convert the polar equation to rectangular form. (b) Use a graphing utility to graph the model for \(a=4\) and \(a=6\) . (c) Find the area of the geographical region between the two curves in part (b).

Finding an Equation of a Hyperbola In Exercises \(41-48,\) find an equation of the hyperbola. Vertices: \((0,2),(6,2)\) Asymptotes: \(y=\frac{2}{3} x\) $$ y=4-\frac{2}{3} x $$

In Exercises 41–48, use the results of Exercises 37–40 to find a set of parametric equations for the line or conic. Hyperbola: vertices: \(( \pm 4,0) ;\) foci : \(( \pm 5,0)\)

Graphing a Polar Equation In Exercises \(43-52,\) use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. $$ r=\frac{2}{1+\cos \theta} $$

Arc Length In Exercises \(45-50\) , find the are length of the curve on the given interval. $$ \text{Parametric Equations} \quad \text{Interval} $$ $$ x=e^{-t} \cos t, \quad y=e^{-t} \sin t \quad 0 \leq t \leq \frac{\pi}{2} $$

Graphing a Polar Equation In Exercises \(43-52,\) use a graphing utility to graph the polar equation. Find an interval for \(\theta\) over which the graph is traced only once. $$ r=\frac{2}{4-3 \sin \theta} $$

Arc Length In Exercises \(45-50\) , find the are length of the curve on the given interval. $$ \text{Parametric Equations} \quad \text{Interval} $$ $$ x=\arcsin t, \quad y=\ln \sqrt{1-t^{2}} \quad 0 \leq t \leq \frac{1}{2} $$