Chapter 6

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

Assume \(t\) is time measured in seconds and velocities have units of \(m / s\) 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. $$v(t)=t^{3}-5 t^{2}+6 t \text { on } 0 \leq t \leq 5$$

Evaluate the following integrals. Include absolute values only when needed. \(\int_{0}^{3} \frac{2 x-1}{x+1} d x\)

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

Verify each identity using the definitions of the hyperbolic functions. \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) (Hint: Begin with the right side of the equation.)

Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. $$\rho(x)=x \sqrt{2-x^{2}} ; \text { for } 0 \leq x \leq 1$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\frac{x^{3}}{3}+\frac{1}{4 x} \text { on }\left[\frac{1}{2}, 2\right]$$

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. $$y=\cos x^{2}, y=0, \text { for } 0 \leq x \leq \sqrt{\pi / 2}$$

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. How long will it take an initial deposit of \(\$ 1500\) to increase in value to \(\$ 2500\) in a saving account with an APY of 3.1 \(\%\) ? Assume the interest rate remains constant and no additional deposits or withdrawals are made.

Assume \(t\) is time measured in seconds and velocities have units of \(m / s\) 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. $$v(t)=50 e^{-2 t} \text { on } 0 \leq t \leq 4$$

Evaluate the following integrals. Include absolute values only when needed. \(\int \tan 10 x d x\)

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

Mass of one-dimensional objects Find the mass of the following thin bars with the given density function. $$\rho(x)=\left\\{\begin{array}{ll} 1 & \text { if } 0 \leq x \leq 2 \\ 2 & \text { if } 2 < x \leq 3 \end{array}\right.$$

Find the area of the surface generated when the given curve is revolved about the \(x\) -axis. $$y=\sqrt{5 x-x^{2}} \text { on }[1,4]$$

Use the general slicing method to find the volume of the following solids. The pyramid with a square base \(4 \mathrm{m}\) on a side and a height of \(2 \mathrm{m}\) (Use calculus.)

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. Between 2005 and \(2010,\) the average rate of inflation was about \(3 \% /\) yr (as measured by the Consumer Price Index). If a cart of groceries cost \(\$ 100\) in 2005 , what will it cost in 2018 , assuming the rate of inflation remains constant?

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=\sin t \text { on }[0,2 \pi] ; s(0)=1$$

Evaluate the following integrals. Include absolute values only when needed. \(\int_{e}^{e^{2}} \frac{d x}{x \ln ^{3} x}\)

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

Verify each identity using the definitions of the hyperbolic functions. $$\cosh x+\sinh x=e^{x}$$

Sketch each region (if a figure is not given) and then find its total area. The region bounded by \(y=\sin x, y=\cos x,\) and the \(x\) -axis between \(x=0\) and \(x=\pi / 2\)

Use the general slicing method to find the volume of the following solids. The tetrahedron (pyramid with four triangular faces), all of whose edges have length 4.

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 number of cells in a tumor doubles every 6 weeks starting with 8 cells. After how many weeks does the tumor have 1500 cells?

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=-t^{3}+3 t^{2}-2 t \text { on }[0,3] ; s(0)=4$$

Evaluate the following integrals. Include absolute values only when needed. \(\int_{0}^{\pi / 2} \frac{\sin x}{1+\cos x} d x\)

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

Use the given identity to verify the related identity. Use the fundamental identity \(\cosh ^{2} x-\sinh ^{2} x=1\) to verify the identity \(\operatorname{coth}^{2} x-1=\operatorname{csch}^{2} x.\)

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume that \(x\) and \(y\) are measured in meters. The spherical zone generated when the upper portion of the circle \(x^{2}+y^{2}=100\) on the interval [-8,8] is revolved about the \(x\) -axis.

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 \(x\) -axis. $$y=8, y=2 x+2, x=0, \text { and } x=2$$

Sketch each region (if a figure is not given) and then find its total area. The regions between \(y=\sin x\) and \(y=\sin 2 x,\) for \(0 \leq x \leq \pi\)

According to the 2010 census, the U.S. population was 309 million with an estimated growth rate of \(0.8 \% / \mathrm{yr}\). a. Based on these figures, find the doubling time and project the population in 2050 . b. Suppose the actual growth rate is just 0.2 percentage point lower than \(0.8 \% / \mathrm{yr}(0.6 \%) .\) What are the resulting doubling time and projected 2050 population? Repeat these calculations assuming the growth rate is 0.2 percentage point higher than \(0.8 \% / \mathrm{yr}\). c. Comment on the sensitivity of these projections to the growth rate.

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=6-2 t \text { on }[0,5] ; s(0)=0$$

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 \(x\) -axis. $$y=4-x, y=2, \text { and } x=0$$

Evaluate the following integrals. Include absolute values only when needed. \(\int \frac{e^{2 x}}{4+e^{2 x}} d x\)

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. $$y=x^{2} \text { on }[-1,1]$$

How much work is required to move an object from \(x=0\) to \(x=3\) (measured in meters) in the presence of a force (in \(\mathrm{N}\) ) given by \(F(x)=2 x\) acting along the \(x\) -axis?

Use the given identity to verify the related identity. Use the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x\) to verify the identities \(\cosh ^{2} x=\frac{\cosh 2 x+1}{2}\) and \(\sinh ^{2} x=\frac{\cosh 2 x-1}{2}.\)

Find the area of the surface generated when the given curve is revolved about the \(y\) -axis. $$y=(3 x)^{1 / 3}, \text { for } 0 \leq x \leq \frac{8}{3}$$

Sketch each region (if a figure is not given) and then find its total area. The region bounded by \(y=x, y=1 / x, y=0,\) and \(x=2\)

On the first day of the year \((t=0),\) a city uses electricity at a rate of \(2000 \mathrm{MW}\). That rate is projected to increase at a rate of \(1.3 \%\) per year. a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city. b. Find the total energy (in \(\mathrm{MW}\) -yr) used by the city over four full years beginning at \(t=0\). c. Find a function that gives the total energy used (in \(\mathrm{MW}\) -yr) between \(t=0\) and any future time \(t > 0\).

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=3 \sin \pi t \text { on }[0,4] ; s(0)=1$$

Let \(R\) be the region bounded by the following curves. Use the disk method to find the volume of the solid generated when \(R\) is revolved about the \(x\) -axis. \(y=2-2 x, y=0, x=0\) (Verify that your answer agrees with the volume formula for a cone.)

Evaluate the following integrals. Include absolute values only when needed. \(\int \frac{d x}{x \ln x \ln (\ln x)}\)

How much work is required to move an object from \(x=1\) to \(x=3\) (measured in meters) in the presence of a force (in \(\mathrm{N}\) ) given by \(F(x)=2 / x^{2}\) acting along the \(x\) -axis?

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. $$y=\sin x \text { on }[0, \pi]$$

Use the given identity to verify the related identity. Use the identity \(\cosh (x+y)=\cosh x \cosh y+\sinh x \sinh y\) to verify the identity \(\cosh 2 x=\cosh ^{2} x+\sinh ^{2} x.\)

Find the area of the surface generated when the given curve is revolved about the \(y\) -axis. $$y=\frac{x^{2}}{4}, \text { for } 2 \leq x \leq 4$$

Sketch each region (if a figure is not given) and then find its total area. The regions in the first quadrant on the interval [0,2] bounded by \(y=4 x-x^{2}\) and \(y=4 x-4\)

Texas had the largest increase in population of any state in the United States from 2000 to 2010 . During that decade, Texas grew from 20.9 million in 2000 to 25.1 million in 2010. Use an exponential growth model to predict the population of Texas in 2025.

Consider an object moving along a line with the following velocities and initial positions. a. Graph the velocity function on the given interval and determine when the object is moving in the positive direction and when it is moving in the negative direction. b. Determine the position function, for \(t \geq 0,\) using both the antiderivative method and the Fundamental Theorem of Calculus (Theorem 6.1 ). Check for agreement between the two methods. c. Graph the position function on the given interval. $$v(t)=9-t^{2} \text { on }[0,4] ; s(0)=-2$$