Chapter 6

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=1 /(x+1), y=1-x / 3 ;\) revolved about the \(x\) -axis

A quantity grows exponentially according to \(y(t)=y_{0} e^{k t} .\) What is the relationship between \(m, n,\) and \(p\) such that \(y(p)=\sqrt{y(m) y(n)} ?\)

For the following regions \(R\), determine which is greater - the volume of the solid generated when \(R\) is revolved about the \(x\) -axis or about the y-axis. \(R\) is bounded by \(y=x^{2}\) and \(y=\sqrt{8 x}\)

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=(x-2)^{3}-2, x=0,\) and \(y=25 ;\) revolved about the \(y\) -axis

Use the most efficient strategy for computing the area of the following regions. The region bounded by \(x=y(y-1)\) and \(x=-y(y-1)\)

The same exponential growth function can be written in the forms \(y(t)=y_{0} e^{k t}, y(t)=y_{0}(1+r)^{t}\) and \(y(t)=y_{0} 2^{t / T_{2}} .\) Write \(k\) as a function \(r, r\) as a function of \(T_{2}\) and \(T_{2}\) as a function of \(k\)

Determine whether the following statements are true and give an explanation or counterexample. a. A pyramid is a solid of revolution. b. The volume of a hemisphere can be computed using the disk method. c. Let \(R_{1}\) be the region bounded by \(y=\cos x\) and the \(x\) -axis on \([-\pi / 2, \pi / 2] .\) Let \(R_{2}\) be the region bounded by \(y=\sin x\) and the \(x\) -axis on \([0, \pi] .\) The volumes of the solids generated when \(R_{1}\) and \(R_{2}\) are revolved about the \(x\) -axis are equal.

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=\sqrt{\ln x}, y=\sqrt{\ln x^{2}},\) and \(y=1 ;\) revolved about the \(x\) -axis

Use the most efficient strategy for computing the area of the following regions. The region bounded by \(x=y(y-1)\) and \(y=x / 3\)

a. Sketch the graphs of the functions f and g and find the \(x\) -coordinate of the points at which they intersect. b. Compute the area of the region described. \(f(x)=\operatorname{sech} x, g(x)=\tanh x ;\) the region bounded by the graphs of \(f, g,\) and the \(y\) -axis

A diving pool that is 4 m deep and full of water has a viewing window on one of its vertical walls. Find the force on the following windows. The window is a circle, with a radius of \(0.5 \mathrm{m}\), tangent to the bottom of the pool.

Use a calculator to make a table similar to Table 2 to approximate the following limits. Confirm your result with l'Hôpital's Rule. $$\lim _{x \rightarrow 0} \frac{\ln (1+x)}{x}$$

Define the relative growth rate of the function \(f\) over the time interval \(T\) to be the relative change in \(f\) over an interval of length \(T\) : $$R_{T}=\frac{f(t+T)-f(t)}{f(t)}$$ Show that for the exponential function \(y(t)=y_{0} e^{k t},\) the relative growth rate \(R_{T}\) is constant for any \(T ;\) that is, choose any \(T\) and show that \(R_{T}\) is constant for all \(t\)

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=(\ln x) / \sqrt{x}, y=0,\) and \(x=2\) revolved about the \(x\) -axis

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=6 /(x+3), y=2-x ;\) revolved about the \(x\) -axis

Use the most efficient strategy for computing the area of the following regions. The region bounded by \(y=x^{3}, y=-x^{3},\) and \(3 y-7 x-10=0\)

Zero net area Consider the function \(f(x)=\frac{1-x}{x}\) a. Are there numbers \(01\) such that \(\int_{1 / a}^{a} f(x) d x=0 ?\)

Evaluate the following derivatives. \(f(x)=\cosh ^{-1} 4 x\)

Determine whether the following statements are true and give an explanation or counterexample. a. The distance traveled by an object moving along a line is the same as the displacement of the object. b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal. c. Consider a tank that is filled and drained at a flow rate of \(V^{\prime}(t)=1-t^{2} / 100(\mathrm{gal} / \mathrm{min}),\) for \(t \geq 0 .\) It follows that the volume of water in the tank increases for 10 min and then decreases until the tank is empty. d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from \(A\) units to \(2 A\) units is greater than the cost of increasing production from \(2 A\) units to \(3 A\) units.

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=1 / \sqrt{x}, y=0, x=2,\) and \(x=6\) revolved about the \(x\) -axis

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=x-x^{4}, y=0 ;\) revolved around the \(x\) -axis

Use the most efficient strategy for computing the area of the following regions. The region bounded by \(y=\sqrt{x}, y=2 x-15,\) and \(y=0\)

Two bars of length \(L\) have densities \(\rho_{1}(x)=4 e^{-x}\) and \(\rho_{2}(x)=6 e^{-2 x},\) for \(0 \leq x \leq L\) a. For what values of \(L\) is bar 1 heavier than bar \(2 ?\) b. As the lengths of the bars increase, do their masses increase without bound? Explain.

Behavior at the origin Using calculus and accurate sketches, explain how the graphs of \(f(x)=x^{p} \ln x\) differ as \(x \rightarrow 0\) for \(p=\frac{1}{2}, 1,\) and 2

Evaluate the following derivatives. \(f(t)=2 \tanh ^{-1} \sqrt{t}\)

Let \(R\) be the region bounded by the following curves. Let \(S\) be the solid generated when \(R\) is revolved about the given axis. If possible, find the volume of \(S\) by both the disk/washer and shell methods. Check that your results agree and state which method is easiest to apply. \(y=x-x^{4}, y=0 ;\) revolved around the \(y\) -axis

Use the most efficient strategy for computing the area of the following regions. The region bounded by \(y=x^{2}-4,4 y-5 x-5=0,\) and \(y=0,\) for \(y \geq 0\)

Hooke's law is applicable to idealized (linear) springs that are not stretched or compressed too far. Consider a nonlinear spring whose restoring force is given by \(F(x)=16 x-0.1 x^{3},\) for \(|x| \leq 7\) a. Graph the restoring force and interpret it. b. How much work is done in stretching the spring from its equilibrium position \((x=0)\) to \(x=1.5 ?\) c. How much work is done in compressing the spring from its equilibrium position \((x=0)\) to \(x=-2 ?\)

Average value What is the average value of \(f(x)=1 / x\) on the interval \([1, p]\) for \(p>1 ?\) What is the average value of \(f\) as \(p \rightarrow \infty ?\)

Evaluate the following derivatives. \(f(v)=\sinh ^{-1} v^{2}\)

Determine whether the following statements are true and give an explanation or counterexample. a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution. b. If a region is revolved about the \(y\) -axis, then the shell method must be used. c. If a region is revolved about the \(x\) -axis, then in principle it is possible to use the disk/washer method and integrate with respect to \(x\) or the shell method and integrate with respect to \(y\).

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=e^{x}, y=0, x=0,\) and \(x=2\) revolved about the \(x\) -axis

Use the most efficient strategy for computing the area of the following regions. The region in the first quadrant bounded by \(y=\frac{5}{2}-\frac{1}{x}\) and \(y=x\)

A 10-kg mass is attached to a spring that hangs vertically and is stretched 2 m from the equilibrium position of the spring. Assume a linear spring with \(F(x)=k x\) a. How much work is required to compress the spring and lift the mass 0.5 m? b. How much work is required to stretch the spring and lower the mass 0.5 m?

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{2 x}\right)$$

Evaluate the following derivatives. \(f(x)=\operatorname{csch}^{-1}(2 / x)\)

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=e^{-x}, y=e^{x}, x=0,\) and \(x=\ln 4\) revolved about the \(x\) -axis

Use the most efficient strategy for computing the area of the following regions. The region in the first quadrant bounded by \(y=x^{-1}, y=4 x,\) and \(y=x / 4\)

A glass has circular cross sections that taper (linearly) from a radius of \(5 \mathrm{cm}\) at the top of the glass to a radius of \(4 \mathrm{cm}\) at the bottom. The glass is \(15 \mathrm{cm}\) high and full of orange juice. How much work is required to drink all the juice through a straw if your mouth is \(5 \mathrm{cm}\) above the top of the glass? Assume the density of orange juice equals the density of water.

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(e^{-10 x^{2}}\right)$$

Evaluate the following derivatives. \(f(x)=x \sinh ^{-1} x-\sqrt{x^{2}+1}\)

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=\ln x, y=\ln x^{2},\) and \(y=\ln 8\) revolved about the \(y\) -axis

Let \(f(x)=x^{p}\) and \(g(x)=x^{1 / q},\) where \(p>1\) and \(q>1\) are positive integers. Let \(R_{1}\) be the region in the first quadrant between \(y=f(x)\) and \(y=x\) and let \(R_{2}\) be the region in the first quadrant between \(y=g(x)\) and \(y=x\) a. Find the area of \(R_{1}\) and \(R_{2}\) when \(p=q,\) and determine which region has the greater area. b. Find the area of \(R_{1}\) and \(R_{2}\) when \(p>q\), and determine which region has the greater area. c. Find the area of \(R_{1}\) and \(R_{2}\) when \(p

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left(x^{\tan x}\right)$$

Evaluate the following derivatives. \(f(u)=\sinh ^{-1}(\tan u)\)

Consider the following velocity functions. In each case, complete the sentence: The same distance could have been traveled over the given time period at a constant velocity of _____. $$v(t)=2 \sin t, \text { for } 0 \leq t \leq \pi$$

Find the volume of the following solids of revolution. Sketch the region in question. The region bounded by \(y=1 /\left(x^{2}+1\right), y=0, x=1,\) and \(x=4\) revolved about the \(y\) -axis

Find the volume of the solid of revolution. Sketch the region in question. The region bounded by \(y=e^{-x}, y=0, x=0,\) and \(x=p > 0\) revolved about the \(x\) -axis (Is the volume bounded as \(p \rightarrow \infty ?\) )

For large distances from the surface of Earth, the gravitational force is given by \(F(x)=G M m /(x+R)^{2},\) where \(G=6.7 \times 10^{-11} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{kg}^{2}\) is the gravitational constant, \(M=6 \times 10^{24} \mathrm{kg}\) is the mass of Earth, \(m\) is the mass of the object in the gravitational field, \(R=6.378 \times 10^{6} \mathrm{m}\) is the radius of Earth, and \(x \geq 0\) is the distance above the surface of Earth (in meters). a. How much work is required to launch a rocket with a mass of \(500 \mathrm{kg}\) in a vertical flight path to a height of \(2500 \mathrm{km}\) (from Earth's surface)? b. Find the work required to launch the rocket to a height of \(x\) kilometers, for \(x>0\) c. How much work is required to reach outer space \((x \rightarrow \infty) ?\) d. Equate the work in part (c) to the initial kinetic energy of the rocket, \(\frac{1}{2} m v^{2},\) to compute the escape velocity of the rocket.

Compute the following derivatives using the method of your choice. $$\frac{d}{d x}\left[\left(\frac{1}{x}\right)^{x}\right]$$