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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises sketch the region bounded by the graphs of the equations and find the area of the region. y = x, y = x³
In Exercises sketch the region bounded by the graphs of the equations and find the area of the region. y sin x, y = cos x, 4 ≤x≤ 5 T 4
In Exercises sketch the region bounded by the graphs of the equations and find the area of the region. x = cos y, X = 1 2' 3 F|M T ≤ y ≤ 7π 3
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations, and use the integration capabilities of the graphing utility to find the area of the region. y = x² - 4x + 3, y = x³, x = 0
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations, and use the integration capabilities of the graphing utility to find the area of the region. y = x² = 2x², y = 2x²
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations, and use the integration capabilities of the graphing utility to find the area of the region. y = x² 8x + 3, y = 3 + 8x - x² -
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations, and use the integration capabilities of the graphing utility to find the area of the region. √x + √√√y = 1, y = 0, x = 0
In Exercises use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given line(s).(a) The x-axis(b) The y-axis(c) The line x = 3(d) The line x = 6 y = x, y = 0, x = 3
Estimate the surface area of the pond using (a) The Trapezoidal Rule (b) Simpson’s Rule I I 50 ft I I I I I I 54 ft I I 82 ft I I I 1 I I I I 82 ft I I I I I 73 ft I I I I I I I 20 ft I 75 ft I I I I I 1 1 1 80 ft 1
In Exercises use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given line(s).revolved about the x-axis y 1 √1+x²² y = 0, x= -1, x = 1
In Exercises use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given line(s).revolved about the y-axis y 1 y = 0, x = 2, x = 5
In Exercises use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given line(s).(a) The x-axis(b) The line y = 2(c) The y-axis (d) The line x = -1 y = √x, y = 2, x = 0
In Exercises use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given line(s).revolved about the y-axis y 1 x4 + 1' y = 0, x = 0, x = 1
In Exercises use the disk method or the shell method to find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given line(s).revolved about the x-axis y = ex, y = 0, x=0, x= 1
The models R₁ = 6.4 +0.2t + 0.01t² and R₂ = 8.4 +0.35t give the revenue (in billions of dollars) for a large corporation. Both models are estimates of the revenues from 2015 through 2020, with t = 15 corresponding to 2015. Which model projects the greater revenue? How much more total revenue
In Exercises find the arc length of the graph of the function over the indicated interval. 4 f(x) = x³/4, [0,4]
In Exercises find the arc length of the graph of the function over the indicated interval. y = x² + 2₁ [1.3] 2x'
A gasoline tank is an oblate spheroid generated by revolving the region bounded by the graph of about the y-axis, where and are measured in feet. Find the depth of the gasoline in the tank when it is filled to one-fourth its capacity. x² + 16 9 || 1
A cable of a suspension bridge forms a catenary modeled by the equationwhere x and y are measured in feet. Use the integration capabilities of a graphing utility to approximate the length of the cable. y = 300 cosh X 2000 - 280, - 2000 ≤ x ≤ 2000
Determine which value best approximates the length of the arc represented by the integral(Make your selection on the basis of a sketch of the arc and not by performing any calculations.)(a) 10 (b) -5(c) 2(d) 4 (e) 1 2 d 4 S [√ ₁ + [ á ( + 1)]³ dx. 1 dx x
Find the volume of the solid whose base is bounded by the circle x² + y² = 9 and the cross sections perpendicular to the x-axis are equilateral triangles.
Find the work done by the force F shown in the figure. Pounds 12 10 8 t 2 F -(9, 4). |||| x 2 4 6 8 10 12 Feet
Use integration to find the lateral surface area of a right circular cone of height 4 and radius 3.
The region bounded by the graphs of y = 2√x, y = 0, x = 3, and x = 8 is revolved about the x-axis. Find the surface area of the solid generated.
Find the center of mass of the point masses lying on the x-axis. mı = 8, m = 12, m = 6, m = 14 = x₁ = -1, x₂ = 2, X3 = 5, X₁ = 7
A force of 5 pounds is needed to stretch a spring 1 inch from its natural position. Find the work done in stretching the spring from its natural length of 10 inches to a length of 15 inches.
A force of 50 pounds is needed to stretch a spring 1 inch from its natural position. Find the work done in stretching the spring from its natural length of 10 inches to double that length.
A water well has an 8-inch casing (diameter) and is 190 feet deep. The water is 25 feet from the top of the well. Determine the amount of work done in pumping the well dry, assuming that no water enters it while it is being pumped.
A quantity of gas with an initial volume of 2 cubic feet and a pressure of 800 pounds per square foot expands to a volume of 3 cubic feet. Find the work done by the gas. Assume that the pressure is inversely proportional to the volume.
Find the center of mass of the given system of point masses. mi 3 2 (x, y) (2, 1) (-3,2) 6 (4, -1) 9 (6,5)
A chain 10 feet long weighs 4 pounds per foot and is hung from a platform 20 feet above the ground. How much work is required to raise the entire chain to the 20-foot level?
In Exercises find the centroid of the region bounded by the graphs of the equations. y = x², y = 2x + 3
In Exercises find the centroid of the region bounded by the graphs of the equations. y = x²/³, y = x
A blade on an industrial fan has the configuration of a semicircle attached to a trapezoid (see figure). Find the centroid of the blade. 432 1 - 1 -2 -3 -4 y 1 2 3 4 5 7
A windlass, 200 feet above ground level on the top of a building, uses a cable weighing 5 pounds per foot. Find the work done in winding up the cable when(a) One end is at ground level.(b) There is a 300-pound load attached to the end of the cable.
The work done by a variable force in a press is 80 foot-pounds. The press moves a distance of 4 feet, and the force is a quadratic of the form F = ax². Find a.
Find the fluid force on the vertical plate submerged in seawater (see figure). 6 ft 3 ft 4 ft-
The figure is the vertical side of a form for poured concrete that weights 140.7 pounds per cubic foot. Determine the force on this part of the concrete form. 7 ft- 5 ft
Use the Theorem of Pappus to find thevolume of the torus formed by revolving the circle(x - 4)² + y² = 4 about the y-axis.
A swimming pool is 5 feet deep at one end and 10 feet deep at the other, and the bottom is an inclined plane. The length and width of the pool are 40 feet and 20 feet. If the pool is full of water, what is the fluid force on each of the vertical walls?
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = x, y = 0, x = 2
Explain why the answer in part (b) of Exercise 15 is not twice the answer in part (a).Data from in Exercise 15A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (see figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out
In Exercises find the center of mass of the point masses lying on the x-axis. m₁ = 7, m₂ = 4, m3 = 3, m₁ = 8 X₁ = −3, X₂ = -2, x3 = 5, x₁ = 4
In Exercises find the center of mass of the point masses lying on the x-axis. m₁ = 8, m₂ = 5, m3 = 5, m₁ = 12, m5 = 2 ms 0, x₁ X3 x4 X5 = 3, x X₁ = -2, x₂ = 6, x3 = = -5
In Exercises find the center of mass of the point masses lying on the x-axis. mı = 7, m X₁ = 3, m = 5 ▬ 5, x = 0, 13 = 3
(a) Translate each point mass in Exercise 3 to the right four units and determine the resulting center of mass.(b) Translate each point mass in Exercise 4 to the left two units and determine the resulting center of mass.Data from in Exercise 3-4In Exercises find the center of mass of the point
In Exercises find the center of mass of the point masses lying on the x-axis. m₁ = 1, m₂ = 1, m₂ = 3, m3 = 2, m4 = 9, M5 = 5 x₁ = 6, x₂ = 10, x3 = 3, x4 = 2, x₂ = 4 X3 X5
In Exercises consider a beam of length L with a fulcrum x feet from one end (see figure). There are objects with weights W1 and W2 placed on opposite ends of the beam. Find x such that the system is in equilibrium.Two children weighing 48 pounds and 72 pounds are going to play on a seesaw that is
In Exercises consider a beam of length L with a fulcrum x feet from one end (see figure). There are objects with weights W1 and W2 placed on opposite ends of the beam. Find x such that the system is in equilibrium.In order to move a 600-pound rock, a person weighing 200 pounds wants to balance it
In Exercises find the center of mass of the given system of point masses. m; (x, y) 10 (1,-1) 2 (5,5) 5 (-4,0)
In Exercises find the center of mass of the given system of point masses. m; (x, y) 12 6 4.5 15 (2, 3) (-1,5) (6,8) (2,-2)
In Exercises find the center of mass of the given system of point masses. m; (x, y) 5 (2, 2) 1 3 (-3, 1) (1,-4)
Use the result of Exercise 5 to make a conjecture about the change in the center of mass that results when each point mass is translated k units horizontally.Data from in Exercise 5(a) Translate each point mass in Exercise 3 to the right four units and determine the resulting center of mass.(b)
In Exercises find the center of mass of the given system of point masses. m; (x, y) 3 (-2,-3) 4 (5,5) 2 1 (7,1) (0, 0) 6 (-3,0)
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = 6 x, y = 0, x = 0 -
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = x², y = x³
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = √√√x, y = 0, x = 4
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = -x² + 4x + 2, y = x + 2
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = x², y = 0, x = 2
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = √√√x + 1, y = x + 1
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = x2/3, y = 4 =
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = √√x, y = x
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = 4y², x = 0
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = 3y - y², x = 0
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. y = x²/3, y = 0, x = 8
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = -y, x = 2y - y²
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y = xe-x/2. ², y = = 0, x = 0, x = 4
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y = 53/400 x², y = 0
In Exercises find Mx, My, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = y + 2, x = y²
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y= 8 x² + 4 y = 0, x= -2, x = 2
In Exercises use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y = 10x 125 x³, y = 0
In Exercises introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.) 2 -1 -2- о ←1→
In Exercises introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.) 2 TAL> ←1+ -21 2
In Exercises introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.) 2 3 7 1 2 4 4 H 1 3 5
In Exercises introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.) [² 710 12- 6
Find the center of mass of the lamina in Exercise 31 when the circular portion of the lamina has twice the density of the square portion of the lamina.Data from in Exercise 31In Exercises introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina.
In Exercises use the Theorem of Pappus to find the volume of the solid of revolution.Data from Theorem of PappusThe torus formed by revolving the circleabout the y-axis THEOREM 7.1 The Theorem of Pappus Let R be a region in a plane and let L be a line in the same plane such that L does not
In Exercises use the Theorem of Pappus to find the volume of the solid of revolution.Data from Theorem of PappusThe torus formed by revolving the circleabout the x-axis THEOREM 7.1 The Theorem of Pappus Let R be a region in a plane and let L be a line in the same plane such that L does not
In Exercises use the Theorem of Pappus to find the volume of the solid of revolution.Data from Theorem of PappusThe solid formed by revolving the region bounded by the graphs of y = x, y = 4, and x = 0 about the x-axis THEOREM 7.1 The Theorem of Pappus Let R be a region in a plane and let L be a
Find the center of mass of the lamina in Exercise 31 when the square portion of the lamina has twice the density of the circular portion of the lamina.Data from in Exercise 31In Exercises introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina.
State the Theorem of Pappus.Data from in Theorem of Pappus THEOREM 7.1 The Theorem of Pappus Let R be a region in a plane and let L be a line in the same plane such that L does not intersect the interior of R, as shown in Figure 7.63. If r is the distance between the centroid of R and the line,
In Exercises use the Theorem of Pappus to find the volume of the solid of revolution.Data from Theorem of PappusThe solid formed by revolving the region bounded by the graphs of y = 2√x - 2, y = 0, and x = 6 about the y-axis THEOREM 7.1 The Theorem of Pappus Let R be a region in a plane and let L
The centroid of the plane region bounded by the graphs of y = ƒ(x),y = 0, x = 0, and x = 3 is (1.2, 1.4). Is it possibleto find the centroid of each of the regions boundedby the graphs of the following sets of equations? Ifso, identify the centroid and explain your answer.(a)(b)(c)(d)
In Exercises find and/or verify the centroid of the common region used in engineering.Show that the centroid of the triangle with vertices (-a, 0), (a, 0), and (b, c) is the point of intersection of the medians (see figure). (-a, 0) y (b, c) (a,0)
In Exercises find and/or verify the centroid of the common region used in engineering.Show that the centroid of the parallelogram with vertices (0, 0), (a, 0), (b, c), and (a + b, c) is the point of intersection of the diagonals (see figure). y (b, c) (a + b, c) (a,0) X
In Exercises find and/or verify the centroid of the common region used in engineering.Find the centroid of the trapezoid with vertices (0, 0), (0, a), (c, b), and (c, 0). Show that it is the intersection of the line connecting the midpoints of the parallel sides and the line connecting the extended
In Exercises find and/or verify the centroid of the common region used in engineering.Find the centroid of the region bounded by the graphs of y = √² - x² and y = 0 (see figure). -r y
In Exercises find and/or verify the centroid of the common region used in engineering.Find the centroid of the region bounded by b the graphs of y = b/a √a² - x² and y = 0 (see figure). -a y b D X
In Exercises find and/or verify the centroid of the common region used in engineering.Find the centroid of the parabolic spandrel shown in the figure. y Parabolic spandrel (1, 1) y=2x-x² (0, 0) X
Consider the graphs of y = x² and y = b, where b > 0.(a) Sketch a graph of the region.(b) Use the graph in part (a) to determine x̅. Explain.(c) Set up the integral for finding My. Because of the form ofthe integrand, the value of the integral can be obtainedwithout integrating. What is
Let the point masses m₁, m₂,..., mn, be located at (x₁, y₁), (x₂, y₂),..., (xn, yn). Define the center of mass (x̅, y̅).
What is a planar lamina? Describe what is meant by the center of mass (x̅, y̅) of a planar lamina.
The manufacturer of glass for a window in a conversion van needs to approximate its center of mass. A coordinate system is superimposed on a prototype of the glass (see figure). The measurements (in centimeters) for the right half of the symmetric piece of glass are listed in the table.(a) Use
The manufacturer of a boat needs to approximate the center of mass of a section of the hull. A coordinate system is superimposed on a prototype (see figure). The measurements (in feet) for the right half of the symmetric prototype are listed in the table.(a) Use Simpson’s Rule to approximate the
In Exercises use the Second Theorem of Pappus, which is stated as follows. If a segment of a plane curve C is revolved about an axis that does not intersect the curve (except possibly at its endpoints), the area S of the resulting surface of revolution is equal to the product of the length of C
In Exercises use the Second Theorem of Pappus, which is stated as follows. If a segment of a plane curve C is revolved about an axis that does not intersect the curve (except possibly at its endpoints), the area S of the resulting surface of revolution is equal to the product of the length of C
Let n ≥ 1 be constant, and consider the region bounded by ƒ(x) = xn, the x-axis, and x = 1. Findthe centroid of this region. As n → ∞, what does the regionlook like, and where is its centroid?
Let V be the region in the cartesian plane consisting of all points (x, y) satisfying the simultaneous conditions |x| ≤ y ≤ |x| + 3 and y ≤ 4. Find the centroid (x̅, y̅) of V.
In Exercises determine the work done by the constant force.A 1200-pound steel beam is lifted 40 feet.
In Exercises determine the work done by the constant force.An electric hoist lifts a 2500-pound car 6 feet.

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