Chapter 10

The planet Pluto travels in an elliptical orbit around the sun (at one focus). The length of the major axis is \(1.18 \times 10^{10} \mathrm{km}\) and the length of the minor axis is \(1.14 \times 10^{10} \mathrm{km} .\) Use Simp- son's Rule with \(n=10\) to estimate the distance traveled by the planet during one complete orbit around the sun.

Find the exact area of the surface obtained by rotating the given curve about the \(x\) -axis. $$x=3 t-t^{3}, \quad y=3 t^{2}, \quad 0 \leq t \leqslant 1$$

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\) . $$r=\cos (\theta / 3), \quad \theta=\pi$$

Find the exact area of the surface obtained by rotating the given curve about the \(x\) -axis. $$x=a \cos ^{3} \theta, \quad y=a \sin ^{3} \theta, \quad 0 \leqslant \theta \leqslant \pi / 2$$

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\) . $$r=\cos 2 \theta, \quad \theta=\pi / 4$$

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta .\) $$r=1+2 \cos \theta, \quad \theta=\pi / 3$$

If the curve $$x=t+t^{3} \quad y=t-\frac{1}{t^{2}} \quad 1 \leqslant t \leqq 2$$ is rotated about the \(x\) -axis, use your calculator to estimate the area of the resulting surface to three decimal places.

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r=3 \cos \theta$$

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r=1-\sin \theta$$

Find the surface area generated by rotating the given curve about the \(y\) -axis. $$x=3 t^{2}, \quad y=2 t^{3}, \quad 0 \leqq t \leq 5$$

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r=1+\cos \theta$$

Find the surface area generated by rotating the given curve about the \(y\) -axis. $$x=e^{t}-t, \quad y=4 e^{t / 2}, \quad 0 \leqslant t \leqslant 1$$

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r=e^{\theta}$$

If \(f^{\prime}\) is continuous and \(f^{\prime}(t) \neq 0\) for \(a \leqslant t \leqslant b,\) show that the parametric curve \(x=f(t), y=g(t), a \leq t \leq b,\) can be put in the form \(y=F(x) .\left[\) Hint. Show that \(f^{-1}\) exists. \right\(]\)

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r=2+\sin \theta$$

\(63-68\) Find the points on the given curve where the tangent line is horizontal or vertical. $$r^{2}=\sin 2 \theta$$

The curvature at a point \(P\) of a curve is defined as $$\kappa=\left|\frac{d \phi}{d s}\right|$$ where \(\phi\) is the angle of inclination of the tangent line at \(P\) as shown in the figure. Thus the curvature is the absolute value of the rate of change of \(\phi\) with respect to arc length. It can be regarded as a measure of the rate of change of direc- tion of the curve at \(P\) and will be studied in greater detail in Chapter \(13 .\) (a) For a parametric curve \(x=x(t), y=y(t),\) derive the formula $$\kappa=\frac{|\dot{x} y-x y|}{\left[\dot{x}^{2}+\dot{y}^{2}\right]^{3 / 2}}$$ where the dots indicate derivatives with respect to \(t,\) so \(\dot{x}=d x / d t .\left[\)Hint. Use \(\phi=\tan ^{-1}(d y / d x)\) and Formula 2\right. to find \(d \phi / d t\) . Then use the Chain Rule to find \(d \phi / d s\) . (b) By regarding a curve \(y=f(x)\) as the parametric curve \(x=x, y=f(x),\) with parameter \(x,\) show that the formula in part (a) becomes $$\kappa=\frac{\left|d^{2} y / d x^{2}\right|}{\left[1+(d y / d x)^{2}\right]^{3 / 2}}$$

Show that the polar equation \(r=a \sin \theta+b \cos \theta,\) where \(a b \neq 0,\) represents a circle, and find its center and radius.

Show that the curves \(r=a \sin \theta\) and \(r=a \cos \theta\) intersect at right angles.

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=1+2 \sin (\theta / 2) \quad$$ (nephroid of Freeth)

(a) Show that the curvature at each point of a straight line is \(\kappa=0 .\) (b) Show that the curvature at each point of a circle of radius \(r\) is \(\kappa=1 / r=1 / r\).

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=\sqrt{1-0.8 \sin ^{2} \theta} \quad$$ (hippopede)

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=e^{\sin \theta}-2 \cos (4 \theta) \quad$$ (butterfly curve)

A string is wound around a circle and then unwound while being held taut. The curve traced by the point \(P\) at the end of the string is called the involute of the circle. If the circle has radius \(r\) and center \(O\) and the initial position of \(P\) is \((r, 0),\) and if the parameter \(\theta\) is chosen as in the figure, show that parametric equations of the involute are $$x=r(\cos \theta+\theta \sin \theta) \quad y=r(\sin \theta-\theta \cos \theta)$$

A cow is tied to a silo with radius \(r\) by a rope just long enough to reach the opposite side of the silo. Find the area available for grazing by the cow.

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=\sin ^{2}(4 \theta)+\cos (4 \theta)$$

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=2-5 \sin (\theta / 6)$$

\(71-76\) Use a graphing device to graph the polar curve. Choose the parameter interval to make sure that you produce the entire curve. $$r=\cos (\theta / 2)+\cos (\theta / 3)$$

How are the graphs of \(r=1+\sin (\theta-\pi / 6)\) and \(r=1+\sin (\theta-\pi / 3)\) related to the graph of \(r=1+\sin \theta\) ? In general, how is the graph of \(r=f(\theta-\alpha)\) related to the graph of \(r=f(\theta) ?\)

(a) Investigate the family of curves defined by the polar equations \(r=\sin n \theta,\) where \(n\) is a positive integer. How is the number of loops related to \(n ?\) (b) What happens if the equation in part (a) is replaced by \(r=|\sin n \theta| ?\)

A family of curves is given by the equations \(r=1+c \sin n \theta\) where \(c\) is a real number and \(n\) is a positive integer. How does the graph change as \(n\) increases? How does it change as \(c\) changes? Illustrate by graphing enough members of the family to support your conclusions.

A family of curves has polar equations $$r=\frac{1-a \cos \theta}{1+a \cos \theta}$$ Investigate how the graph changes as the number a changes. In particular, you should identify the transitional values of a for which the basic shape of the curve changes.

The astronomer Giovanni Cassini \((1625-1712)\) studied the family of curves with polar equations $$r^{4}-2 c^{2} r^{2} \cos 2 \theta+c^{4}-a^{4}=0$$ where \(a\) and \(c\) are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values of a and \(c .\) (Cassini thought that these curves might represent planetary orbits better than Kepler's ellipses.) Investigate the variety of shapes that these curves may have. In particular, how are \(a\) and \(c\) related to each other when the curve splits into two parts?

Let \(P\) be any point (except the origin) on the curve \(r=f(\theta)\) . If \(\psi\) is the angle between the tangent line at \(P\) and the radial line \(O P\) , show that $$\tan \psi=\frac{r}{d r / d \theta}$$ [Hint: Observe that \(\psi=\phi-\theta\) in the figure. \(]\)