7.E: Applications of Integration (Exercises)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)7.1: Area Between Curves
Terms and Concepts
1. T/F: The are between curves is always positive.
2. T/F: Calculus can be used to find the area of basic geometric shapes.
3. In your own words, describe how to find the total area enclosed by \(y=f(x)\text{ and }y=g(x)\).
Problems
In Exercises 4-10, find the area of the shaded region in the given graph.
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In Exercises 11-16, find the total area enclosed by the functions \(f\) and \(g\).
11. \(f(x)=2x^2+5x-3,\, g(x)=x^2+4x-1\)
12. \(f(x)=x^2-3x+2,\, g(x)=-3x+3\)
13. \(f(x)=\sin x,\, g(x)=2x/\pi\)
14.. \(f(x)=x^3-4x^2+x-1,\, g(x)=-x^2+2x-4\)
15. \(f(x)=x,\, g(x)=\sqrt{x}\)
16. \(f(x)=-x^3+5x^2+2x+1,\, g(x)=3x^2+x+3\)
17. The functions \(f(x)=\cos (2x)\text{ and }g(x) =\sin x\) intersect infinitely many times, forming an infinite number of repeated, enclosed regions. Find the areas of these regions.
In Exercises 18-22, find the area of the enclosed region in two ways:
1. by treating the boundaries as functions of x, and
2. by treating the boundaries as functions of y.
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In Exercises 23-26, find the are triangle formed by the given three points.
23. \((1,1),\, (2,3)\text{ and }(3,3)\)
24. \((-1,1),\, (1,3)\text{ and }(2,-1)\)
25. \((1,1),\, (3,3)\text{ and }(3,3)\)
26. \((0,0),\, (2,5)\text{ and }(5,2)\)
27. Use the Trapezoidal Rule to approximate the area of the pictured lake whose lengths, in hundreds of feet, are measured in 100-foot increments.
28. Use Simpson's Rule to approximate the area of the pictured lake whose lengths, in hundreds of feet, are measured in 200-foot increments.
7.2: Volume by Cross-Sectional Area: Disk and Washer Methods
Terms and Concepts
1. T/F: A solid of revolution is formed by revolving a shape around an axis.
2. In your own words, explain how the Disk and Washer Methods are related.
3 Explain how the units of volume are found in the integral of Theorem 54: if \(A(x)\) has units of in\(^2\), how does \(\int A(x)\,dx\) have units of in\(^3\)?
Problems
In Exercises 4-7, a region of the Cartesian plane is shaded. Use the Disk/Washer Method to find the volume of the solid of revolution formed by revolving the region about the x-axis.
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In Exercises 8-11, a region of the Cartesian plane is shaded. Use the Disk/Washer Method to find the volume of the solid of revolution formed by revolving the region about the y-axis.
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(Hint: Integration By Parts will be necessary, twice. First let \(u=\text{arccos}^2 x\), then let \(u=\text{arccos } x\).)
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In Exercises 12-17, a region of the Cartesian plane is described. Use the Disk/Washer Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes.
12. Region bounded by: \(y=\sqrt{x},\,y=0\text{ and }x=1.\)
Rotate about:
(a) the x-axis
(b) \(y=1\)
(c) the y-axis
(d) \(x=1\)
13. Region bounded by: \(y=4-x^2\text{ and }y=0.\)
Rotate about:
(a) the x-axis
(b) \(y=4\)
(c) \(y=-1\)
(d) \(x=2\)
14. The triangle with vertices \((1,1),\,(1,2)\text{ and }(2,1).\)
Rotate about:
(a) the x-axis
(b) \(y=2\)
(c) the y-axis
(d) \(x=1\)
15. Region bounded by: \(y=y=x^2-2x+2,\text{ and }y=2x-1.\)
Rotate about:
(a) the x-axis
(b) \(y=1\)
(c) \(y=5\)
16. Region bounded by: \(y=1/\sqrt{x^2+1},\,x=-1,\,x=1 \text{ and the x-axis}.\)
Rotate about:
(a) the x-axis
(b) \(y=1\)
(c) \(y=-1\)
17. Region bounded by: \(y=2x,\,y=x\text{ and }x=2.\)
Rotate about:
(a) the x-axis
(b) \(y=4\)
(c) the y-axis
(d) \(x=2\)
In Exercises 18-21, a solid is described. Orient the solid along the x-axis such that a cross-sectional area function \(A(x)\) can be obtained, then apply Theorem 54 to find the volume of the solid.
18. A right circular cone with height of 10 and base radius of 5.
19. A skew right circular cone with height of 10 and base radius of 5. (Hint: all cross-sections are circles.)
20. A right triangle cone with height of 10 and whose base is a right, isosceles triangle with side length 4.
21. A solid with length 10 with a rectangular base and triangular top, wherein one end is a square with side length 5 and the other end is a triangle with base and height of 5.
7.3: The Shell Method
Terms and Concepts
1. T/F: A solid of revolution is formed by revolving a shape around an axis.
2. T/F: The Shell Method can only be used when the Washer Method fails.
3. T/F: The Shell Method works by integrating cross-sectional areas of a solid.
4. T/F: When finding the volume of a solid of revolution that was revolved around a vertical axis, the Shell Method integrates with respect to x.
Problems
In Exercises 5-8, a region of the Cartesian plane is shaded. Use the Shell Method to find the volume of the solid of revolution formed by revolving the region about the y-axis.
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In Exercises 9-12, a region of the Cartesian plane is shaded. Use the Shell Method to find the volume of the solid of revolution formed by revolving the region about the x-axis.
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In Exercises 13-18, a region of the Cartesian plane is described. Use the Shell Method to find the volume of the solid of revolution formed by rotating the region about each of the given axes.
13. Region bounded by: \(y=\sqrt{x},\,y=0\text{ and }x=1.\)
Rotate about:
(a) the x-axis
(b) \(y=1\)
(c) the y-axis
(d) \(x=1\)
14. Region bounded by: \(y=4-x^2\text{ and }y=0.\)
Rotate about:
(a) \(x=2\)
(b) \(x=-2\)
(c) the x-axis
(d) \(y=4\)
15. The triangle with vertices \((1,1),\,(1,2)\text{ and }(2,1).\)
Rotate about:
(a) the x-axis
(b) \(x=1\)
(c) the x-axis
(d) \(y=2\)
16. Region bounded by: \(y=y=x^2-2x+2,\text{ and }y=2x-1.\)
Rotate about:
(a) the x-axis
(b) \(x=1\)
(c) \(x=-1\)
17. Region bounded by: \(y=1/\sqrt{x^2+1},\,x=1 \text{ and the x and y-axis}.\)
Rotate about:
(a) the x-axis
(b) \(x=1\)
18. Region bounded by: \(y=2x,\,y=x\text{ and }x=2.\)
Rotate about:
(a) the y-axis
(b) \(x=2\)
(c) the x-axis
(d) \(y=4\)
7.4: Arc Length and Surface Area
Terms and Concepts
1. T/F: The integral formula for computing Arc Length was found by first approximating arc length with straight line segments.
2. T/F: The integral formula for computing Arc Length includes a square-root, meaning the integration is probably easy.
Problems
In Exercises 3-12, find the arc length of the function on the given interval.
3. \(f(x) = x\text{ on }[0,1].\)
4. \(f(x) = \sqrt{8}x\text{ on }[-1,1].\)
5. \(f(x) = \frac{1}{3}x^{3/2}-x^{1/2}\text{ on }[0,1].\)
6. \(f(x) = \frac{1}{12}x^3+\frac{1}{x}\text{ on }[1,4].\)
7. \(f(x) = 2x^{3/2}-\frac{1}{6}\sqrt{x}\text{ on }[0,9].\)
8. \(f(x) = \cosh x \text{ on }[-\ln 2,\ln 2].\)
9. \(f(x) = \frac{1}{2}(e^2+e^{-x})\text{ on }[0,\ln 5].\)
10. \(f(x) = \frac{1}{12}x^5+\frac{1}{5x^3}\text{ on }[0.1,1].\)
11. \(f(x) = \ln \left ( \sin x \right ) \text{ on }[\pi/6,\pi/2].\)
12. \(f(x) = \ln \left ( \cos x \right )\text{ on }[0,\pi/4].\)
In Exercises 13-20, set up the integral to compute the arc length of the function on the given interval. Do no evaluate the integral.
13. \(f(x) = x^2\text{ on }[0,1].\)
14. \(f(x) = x^{10}\text{ on }[0,1].\)
15. \(f(x) = \sqrt{x}\text{ on }[0,1].\)
16. \(f(x) = \ln x \text{ on }[1,e].\)
17. \(f(x) = \sqrt{1-x^2}\text{ on }[-1,1].\)(Note: this describes the top half of a circle with radius 1.)
18. \(f(x) = \sqrt{1-x^2/9} \text{ on }[-3,3]\). (Note: this describes the top half of an eclipse with a major axis of length 6 and a minor axis of length 2.)
19. \(f(x) = \frac{1}{x}\text{ on }[1,2]\).
20. \(f(x) = \sec x\text{ on }[-\pi/4, \pi/4]\).
In Exercises 21-28, use Simpson's Rule, with \(n=4\), to approximate the arc length of the function on the given interval. Note: these are the same problems as in Exercises 13-20.
21. \(f(x) = x^2\text{ on }[0,1].\)
22. \(f(x) = x^{10}\text{ on }[0,1].\)
23. \(f(x) = \sqrt{x}\text{ on }[0,1].\) (Note: \(f'(x)\) is not defined at \(x=0\).)
24. \(f(x) = \ln x \text{ on }[1,e].\)
25. \(f(x) = \sqrt{1-x^2}\text{ on }[-1,1].\) (Note: \(f'(x)\) is not defined at the endpoints.)
26. \(f(x) = \sqrt{1-x^2/9} \text{ on }[-3,3]\). (Note: \(f'(x)\) is not defined at the endpoints.)
27. \(f(x) = \frac{1}{x}\text{ on }[1,2]\).
28. \(f(x) = \sec x\text{ on }[-\pi/4, \pi/4]\).
In Exercises 29-33, find the surface area of the described solid of revolution.
29. The solid formed by revolving \(y=2x \text{ on }[0,1]\) about the x-axis.
30. The solid formed by revolving \(y=x^2 \text{ on }[0,1]\) about the y-axis.
31. The solid formed by revolving \(y=x^3 \text{ on }[0,1]\) about the x-axis.
32. The solid formed by revolving \(y=\sqrt{x} \text{ on }[0,1]\) about the x-axis.
33. The solid formed by revolving \(y=\sqrt{1-x^2} \text{ on }[-1,1]\) about the x-axis.
7.5: Work
Terms and Concepts
1. What are the typical units of work?
2. If a man has a mass of 80kg on Earth, will his mass on the moon be bigger, smaller, or the same?
3. If a woman weights 130 lb on Earth, will her weight on the moon be bigger, smaller, or the same?
Problems
4. A 100 ft rope, weighing 0.1 lb/ft, hangs over the edge of a tall building.
(a) How much work is done pulling the entire rope to the top of the building?
(b) How much rope is pulled in when half of the total work is done?
5. A 50 m rope, with a mass density of 0.2 kg/m, hangs over the edge of a tall building.
(a) How much work is done pulling the entire rope to the top of the building?
(b) How much work is done pulling in the first 20 m?
6. A rope of length \(l\) ft hangs over the edge of tall cliff. (Assume the cliff is taller than the length of the rope.) The rope has a weight density of \(d\) lb/ft.
(a) How much work is done pulling the entire rope to the top of the cliff?
(b) What percentage of the total work is done pulling in the first half of the rope?
(c) How much rope is pulled in when half of the total work is done?
7. A 20 m rope with mass density of 0.5 kg/m hangs over the edge of a 10 m building. How much work is done pulling the rope to the top?
8. A crane lifts a 2000 lb load vertically 30 ft with a 1" cable weighing 1.68 lb/ft.
(a) How much work is done lifting the cable alone?
(b) How much work is done lifting the load alone?
(c) Could one conclude that the work done lifting the cable is negligible compared to the work done lifting the load?
9. A 100 lb bag of sand is lifted uniformly 120 ft in one minute. Sand leaks from the bag at a rate of 1/4 lb/s. What is the total work done in lifting the bag?
10. A box weighing 2 lb lifts 10 lb of sand vertically 50 ft. A crack in the box allows the sand to leak out such that 9 lb of sand is in the box at the end of the trip. Assume the sand leaked out at a uniform rate. What is the total work done in lifting the box and sand?
11. A force of 1000 lb compresses a spring 3 in. How much work is performed in compressing the spring?
12. A force of 2 N stretches a spring 5 cm. How much work is performed in stretching the spring?
13. A force of 50 lb compresses a spring from a natural length of 18 in to 12 in. How much work is performed in compressing the spring?
14. A force of 20 lb stretches a spring from a natural length of 6 in to 8 in. How much work is performed in stretching the spring?
15. A force of 7 N stretches a spring from a natural length of 11 cm to 21 cm. How much work is performed in stretching the spring from a length of 16 cm to 21 cm?
16. A force of \(f\) N stretches a spring \(d\) m from its natural length. How much work is performed in stretching the spring?
17. A 20 lb weight is attached to a spring. The weight rests on the spring, compressing the spring from a natural length of 1 ft to 6 in.
How much work is done in lifting the box 1.5 ft (i.e, the spring will be stretched 1 ft beyond its natural length)?
18. A 20 lb weight is attached to a spring. The weight rests on the spring, compressing the spring from a natural length of 1 ft to 6 in.
How much work is done in lifting the box 6 in (i.e, bringing the spring back to its natural length)?
19. A 5 m tall cylindrical tank with radius of 2 m is filled with 3 m of gasoline, with a mass density of 737.22 kg/m\(^3\). Compute the total work performed in pumping all the gasoline to the top of the tank.
20. A 6 ft cylindrical tank with a radius of 3 ft is filled with water, which has a weight density of 62.4 lb/ft\(^3\). The water is to be pumped to a point 2 ft above the top of the tank.
(a) How much work is performed in pumping all the water from the tank?
(b) How much work is performed in pumping 3 ft of water from the tank?
(c) At what point is 1/2 of the total work done?
21. A gasoline tanker is filled with gasoline with a weight density of 45.93 lb/ft\(^3\). The dispensing value at the base is jammed shut, forcing the operator to empty the tank via pumping the gas to a point 1 ft above the top of the tank. Assume the tank is a perfect cylinder, 20 ft long with a diameter of 7.5 ft. How much work is performed in pumping all the gasoline from the tank?
22. A fuel oil storage tank is 10 ft deep with trapezoidal sides, 5 ft at the top of the 2 ft at the bottom, and is 15 ft wide (see diagram below). Given that fuel oil weighs 55.46 lb/ft\(^3\), find the work performed in pumping all the oil from the tank to a point 3 ft above the top of the tank.
23. A conical tank is 5 m deep with a top radius of 3 m. (This is similar to Example 224.) The tank is filled with pure water, with a mass density of 1000 kg/m\(^3\).
(a) Find the work performed in pumping all the water to the top of the tank.
(b) Find the work performed in pumping the top 2.5 m of water to the top of the tank.
(c) Find the work performed in pumping the top half of the water, by volume, to the top of the tank.
24. A water tank has the shape of a truncated cone, with dimensions given below, and is filled with water with a weight density of 62.4 lb/ft\(^3\). Find the work performed in pumping all water to a point 1 ft above the top of the tank.
25. A water tank has the shape of an inverted pyramid, with dimensions given below, and is filled with water with a mass density of 1000 kg/m\(^3\). Find the work performed in pumping all water to a point 5 m above the top of the tank.
26. A water tank has the shape of a truncated, inverted pyramid, with dimensions given blow, and is filled with water with a mass density of 1000 kg/m\(^3\). Find the work performed in pumping all water to a point 1 m above the top of the tank.
7.6: Fluid Forces
Terms and Concepts
1. State in your own words Pascal's Principle.
2. State in your own words how pressure is different from force.
Problems
In Exercises 3-12, find the fluid force exerted on the given plate, submerged in water with a weight density of 62.4 lb/ft\(^3\).
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In Exercises 13-18, the side of a container is pictured. Find the fluid force exerted on this plate when the container is full of:
1. water, with a weight density of 62.4 lb/ft\(^3\)
2. concrete, with a weight density of 150 lb/ft\(^3\).
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19. How deep must the center of a vertically oriented circular plate with a radius of 1 ft be submerged in water, with a weight density of 62.4 lb/ft\(^3\), for the fluid force on the plate to reach 1,000 lb?
20. How deep must the center of a vertically oriented square plate with a side length of 2 ft be submerged in water, with a weight density of 62.4 lb/ft\(^3\), for the fluid force on the plate to reach 1,000 lb?