Chapter 6

A solid has a circular base and cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?

What is the result of integrating a population growth rate between two times \(t=a\) and \(t=b,\) where \(b>a ?\)

Evaluate the following derivatives. $$\left.\frac{d}{d x}\left(x \ln \left(x^{3}\right)\right)\right|_{x=1}$$

A calculator has a built-in \(\sinh ^{-1} x\) function, but no \(\operatorname{csch}^{-1} x\) function. How do you evaluate \(\operatorname{csch}^{-1} 5\) on such a calculator?

Give two examples of processes that are modeled by exponential growth.

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=8 \sqrt{x} \text { on }[9,20]$$

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). $$y=\frac{\left(x^{2}+2\right)^{3 / 2}}{3} ;[0,1]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=\left(1+x^{2}\right)^{-1}, y=0, x=0, \text { and } x=2$$

Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by the curves \(y=x^{2}\) and \(y=2-x^{2}\) and whose cross sections through the solid perpendicular to the \(x\) -axis are squares

Assume \(t\) is time measured in seconds and velocities have units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. $$v(t)=6-2 t ; 0 \leq t \leq 6$$

Explain why you integrate in the vertical direction (parallel to the acceleration due to gravity) rather than the horizontal direction to find the force on the face of a dam.

Evaluate the following derivatives. $$\frac{d}{d x}(\ln (\ln x))$$

On what interval is the formula \(d / d x\left(\tanh ^{-1} x\right)=1 /\left(x^{2}-1\right)\) valid?

Give two examples of processes that are modeled by exponential decay.

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=x^{3} \text { on }[0,1]$$

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). $$y=\frac{x^{3 / 2}}{3}-x^{1 / 2} ;[4,16]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=6-x, y=0, x=2, \text { and } x=4$$

Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by the semicircle \(y=\sqrt{1-x^{2}}\) and the \(x\) -axis and whose cross sections through the solid perpendicular to the \(x\) -axis are squares

Assume \(t\) is time measured in seconds and velocities have units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. $$v(t)=10 \sin 2 t ; 0 \leq t \leq 2 \pi$$

When evaluating the definite integral \(\int_{6}^{8} \frac{d x}{16-x^{2}},\) why must you choose the antiderivative \(\frac{1}{4} \operatorname{coth}^{-1} \frac{x}{4}\) rather than \(\frac{1}{4} \tanh ^{-1} \frac{x}{4} ?\)

Evaluate the following derivatives. $$\frac{d}{d x}(\sin (\ln x))$$

Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=100+10.5 t, g(t)=100 e^{t / 10}$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=x^{3 / 2}-\frac{x^{1 / 2}}{3} \text { on }[1,2]$$

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). $$y=\frac{x^{4}}{4}+\frac{1}{8 x^{2}} ;[1,2]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. \(y=3 x, y=3,\) and \(x=0\) (Do not use the volume formula for a cone.)

Use the general slicing method to find the volume of the following solids. The solid whose base is the region bounded by the curve \(y=\sqrt{\cos x}\) and the \(x\) -axis and whose cross sections through the solid perpendicular to the \(x\) -axis are isosceles right triangles with a horizontal leg in the \(x y\) -plane and a vertical leg above the \(x\) -axis

Sketch the region and find its area. The region bounded by \(y=2(x+1), y=3(x+1),\) and \(x=4\)

Assume \(t\) is time measured in seconds and velocities have units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. $$v(t)=t^{2}-6 t+8 ; 0 \leq t \leq 5$$

How does the graph of the catenary \(y=a \cosh (x / a)\) change as \(a>0\) increases?

Evaluate the following derivatives. $$\frac{d}{d x}\left(\ln \left(\cos ^{2} x\right)\right)$$

Two functions \(f\) and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant. $$f(t)=2200+400 t, g(t)=400 \cdot 2^{t / 20}$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\sqrt{4 x+6} \text { on }[0,5]$$

Arc length calculations Find the arc length of the following curves on the given interval by integrating with respect to \(x\). $$y=\frac{2}{3} x^{3 / 2}-\frac{1}{2} x^{1 / 2} ;[1,9]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. \(y=1-x^{2}, x=0,\) and \(y=0,\) in the first quadrant

Sketch the region and find its area. The region bounded by \(y=\cos x\) and \(y=\sin x\) between \(x=\pi / 4\) and \(x=5 \pi / 4\)

Assume \(t\) is time measured in seconds and velocities have units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. $$v(t)=-t^{2}+5 t-4 ; 0 \leq t \leq 5$$

Use the general slicing method to find the volume of the following solids. The solid with a circular base of radius 5 whose cross sections perpendicular to the base and parallel to the \(x\) -axis are equilateral triangles

Find the mass of the following thin bars with the given density function. $$\rho(x)=2-x / 2 ; \text { for } 0 \leq x \leq 2$$

Verify each identity using the definitions of the hyperbolic functions. \(\tanh x=\frac{e^{2 x}-1}{e^{2 x}+1}\)

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The population of a town with a 2010 population of 90,000 grows at a rate of \(2.4 \% /\) yr. In what year will the population double its initial value (to \(180,000) ?\)

Evaluate the following derivatives. $$\frac{d}{d x}\left((\ln 2 x)^{-5}\right)$$

Arc length by calculator a. Write and simplify the integral that gives the arc length of the following curves on the given interval. b. If necessary, use technology to evaluate or approximate the integral. $$y=x^{2} ;[-1,1]$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\frac{1}{4}\left(e^{2 x}+e^{-2 x}\right) \text { on }[-2,2]$$

Let \(R\) be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when \(R\) is revolved about the \(y\) -axis. $$y=x^{3}-x^{8}+1, y=1$$

Use the general slicing method to find the volume of the following solids. The solid with a semicircular base of radius 5 whose cross sections perpendicular to the base and parallel to the diameter are squares

Sketch the region and find its area. The region bounded by \(y=e^{x}, y=e^{-2 x},\) and \(x=\ln 4\)

Assume \(t\) is time measured in seconds and velocities have units of \(\mathrm{m} / \mathrm{s}\). a. Graph the velocity function over the given interval. Then determine when the motion is in the positive direction and when it is in the negative direction. b. Find the displacement over the given interval. c. Find the distance traveled over the given interval. $$v(t)=t^{3}-5 t^{2}+6 t ; 0 \leq t \leq 5$$

Find the mass of the following thin bars with the given density function. $$\rho(x)=5 e^{-2 x} ; \text { for } 0 \leq x \leq 4$$

Verify each identity using the definitions of the hyperbolic functions. \(\tanh (-x)=-\tanh x\)

Devise the exponential growth function that fits the given data; then answer the accompanying questions. Be sure to identify the reference point\((t=0)\) and units of time. The population of Clark County, Nevada, was 1.9 million in 2008. Assuming an annual growth rate of 4.5\%/yr, what will the county population be in \(2020 ?\)