Chapter 5

Find the value of \(c\) such that the parabola \(y=c x^{2}\) divides the region bounded by the parabola \(y=\frac{1}{9} x^{2}\), and the lines \(y=2\), and \(x=0\) into two subregions of equal area

Damped Harmonic Motion The equation of motion of a weight attached to a spring and a dashpot damping device is $$ x(t)=-\frac{1}{\sqrt{2}} e^{-4 t} \sinh 2 \sqrt{2} t $$ where \(x(t)\), measured in feet, is the displacement from the equilibrium position of the spring system and \(t\) is measured in seconds. a. Find the initial position and the initial velocity of the weight. b. Plot the graph of \(x(t)\). (equilibrium position)

The base of a wooden wedge is in the form of a semicircle with radius \(a\), and its top is a plane that passes through the diameter of the base and makes a \(45^{\circ}\) angle with the plane of the base. Find the volume of the wedge.

Let \(A(x)\) denote the area of the region in the first quadrant completely enclosed by the graphs of \(f(x)=x^{m}\) and \(g(x)=x^{1 / m}\), where \(m\) is a positive integer. a. Find an expression for \(A(m)\). b. Evaluate \(\lim _{m \rightarrow 1} A(m)\) and \(\lim _{m \rightarrow \infty} A(m) .\) Give a geomet- ric interpretation. c. Verify your observations in part (b) by plotting the graphs of \(f\) and \(g\).

Heat-Seeking Missiles In a test conducted on a heat-seeking Missile \(A\), the target missile \(B\), which is initially at a distance of \(b\) miles from Missile \(A\), is launched vertically upward. Assume that Missile \(A\) travels at a constant speed \(v_{A}\), that Missile \(B\) travels at a constant speed \(v_{B}\left(v_{A}>v_{B}\right)\), and that Missile \(A\), which is launched from the origin, is always pointed at Missile \(B\). Then the trajectory of Missile \(A\) is $$ y=\frac{b}{2}\left[\frac{\left(1-\frac{x}{b}\right)^{1+c}}{1+c}-\frac{\left(1-\frac{x}{b}\right)^{1-c}}{1-c}\right]+\frac{b c}{1-c^{2}} $$ where \(c=v_{B} / v_{A}\). The trajectory of Missile \(A\) is a pursuit curve. a. Find the point at which Missile \(A\) intercepts Missile \(B\). b. Show that $$ \frac{d y}{d x}=-\sinh \left[c \ln \left(1-\frac{x}{b}\right)\right] $$ c. Suppose that \(b=1\) and \(c=\frac{1}{2}\). Show that the distance \(D\) traveled by Missile \(A\) for the intercept is \(1 \frac{1}{3} \mathrm{mi}\). Hint: \(D=\int_{0}^{1} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x\) d. Plot the graph of the trajectory of the heat-seeking missile taking \(b=1\) and \(c=\frac{1}{2}\)

Let \(f(x)=\frac{1}{x^{2}+1}\) and \(g(x)=|x|\). a. Plot the graphs of \(f\) and \(g\) using the viewing window \([-1,1] \times[0,1.5]\). Find the points of intersection of the graphs of \(f\) and \(g\) accurate to three decimal places. b. Use a calculator or computer and the result of part (a) to find the area of the region bounded by the graphs of \(f\) and \(g\).

The minimum-surface-of-revolution problem may be stated as follows: Of all curves joining two fixed points, find the one that, when revolved about the \(x\) -axis, will generate a surface of minimum area. It can be shown that the solution to the problem is a catenary. The resulting surface of revolution is called a catenoid. Suppose a catenary described by the equation $$ y=\cosh x \quad a \leq x \leq b $$ is revolved about the \(x\) -axis. Find the surface area of the resulting catenoid.

The curve with equation \(y^{2}-4 x^{3}+4 x^{4}=0\) is called a piriform. a. Plot the curve using the viewing window \([-1,1] \times[-1,1]\) b. Find the area of the region enclosed by the curve accurate to five decimal places.

Capacity of a Fuel Tank The external fuel tank for a fighter aircraft is \(8 \mathrm{~m}\) long. The areas of the cross sections in square meters measured from the front to the back of the tank at \(1-\mathrm{m}\) intervals are summarized in the following table. $$ \begin{array}{l} \begin{array}{|l|c|c|c|c|c|} \hline \boldsymbol{x} \text { (distance } & & & & & \\ \text { from front) } & 0 & 1 & 2 & 3 & 4 \\ \hline \boldsymbol{A}(\boldsymbol{x}) & 0 & 0.3041 & 0.6206 & 0.8937 & 0.8937 \\\ \hline \end{array}\\\ \begin{array}{|l|c|c|c|c|} \hline \begin{array}{l} x \text { (distance } \\ \text { from front) } \end{array} & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{A}(\boldsymbol{x}) & 0.8937 & 0.6206 & 0.3041 & 0 \\ \hline \end{array} \end{array} $$

The curve with equation \(4 y^{2}-4 x y^{2}-x^{2}-x^{3}=0\) is called a right strophoid. a. Plot the curve using the viewing window \([-1.5,1.5] \times[-0.5,0.5]\) b. Find the area of the region enclosed by the loop of the curve.

Find the centroid of the region under the graph of \(f(x)=\cosh x\) on \([-a, a]\)

The Volume of a Pontoon A pontoon is \(12 \mathrm{ft}\) long. The areas of the cross sections in square feet measured from the blueprint at intervals of \(2 \mathrm{ft}\) from the front to the back of the part of the pontoon that is under the waterline are summarized in the following table. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline A(x) & 0 & 3.82 & 4.78 & 3.24 & 2.64 & 1.80 & 0 \\ \hline \end{array} $$

a. Let \(S\) be a solid bounded by planes that are perpendicular to the \(x\) -axis at \(x=0\) and \(x=h\). If the cross-sectional area of \(S\) at any point \(x\) in \([0, h]\) is \(A(x)\), where \(A\) is a polynomial of degree less than or equal to three, show that the volume of the solid is $$ V=\frac{h}{6}\left[A(0)+4 A\left(\frac{h}{2}\right)+A(h)\right] $$ b. Use the result of part (a) to verify the result of Exercise 54 .

Prove that \(\frac{d}{d x} \cosh u=(\sinh u) \frac{d u}{d x} .\)

In Exercises \(71-74\), determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Two cars start out traveling side by side along a straight road at \(t=0 .\) Twenty seconds later, \(\operatorname{Car} A\) is \(30 \mathrm{ft}\) behind Car \(B\). If \(v_{1}\) and \(v_{2}\) are continuous velocity functions for Car \(A\) and Car \(B\), respectively, where \(v_{1}(t)\) and \(v_{2}(t)\) are measured in feet per second, then $$ \int_{0}^{20} v_{2}(t) d t=\int_{0}^{20} v_{1}(t) d t+30 $$

Prove that \(\frac{d}{d x} \operatorname{csch} u=-(\operatorname{csch} u \operatorname{coth} u) \frac{d u}{d x}\).

In Exercises \(71-74\), determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. Suppose that the acceleration of \(\operatorname{Car} A\) and \(\operatorname{Car} B\) along a straight road are \(a_{1}(t) \mathrm{ft} / \mathrm{sec}^{2}\) and \(a_{2}(t) \mathrm{ft} / \mathrm{sec}^{2}\), respectively, over the time interval \(\left[t_{1}, t_{2}\right]\), where \(a_{1}\) and \(a_{2}\) are continuous functions with \(a_{1}(t) \geq a_{2}(t)\) on \(\left[t_{1}, t_{2}\right] .\) Then at time \(t=t_{2}\), Car \(A\) will be traveling \(\int_{t_{1}}^{t_{2}}\left[a_{1}(t)-a_{2}(t)\right] d t \mathrm{ft} / \mathrm{sec}\) faster than Car \(B\). (Assume that \(t\) is measured in seconds.)

Prove that \(\frac{d}{d x} \operatorname{sech} u=-(\operatorname{sech} u \tanh u) \frac{d u}{d x}\).

Prove that \(\frac{d}{d x} \operatorname{coth} u=-\left(\operatorname{csch}^{2} u\right) \frac{d u}{d x}\).

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. \((\sinh x+\cosh x)^{3}>0\) for all \(x\) in \((-\infty, \infty) .\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. \(\int_{-\pi}^{\pi}(\cos x) \sinh x d x=0\)