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mathematics
calculus 10th edition

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

Evaluate the integral. ye-y/x dy
Use a computer algebra system to approximate the triple iterated integral. √9-y² HD /9-y² Jo y dz dx dy
Sketch the region R and evaluate the iterated integral ∫R∫ ƒ(x,y) dA. SS aJ-√√√√a²-x 22 (x + y) dy dx
Find the area of the surface given by z = ƒ(x, y) over the region R. f(x, y) = In sec x| R = {(x, y): 0 ≤ x ≤ 7,0 ≤ y ≤ tan x =
Evaluate the integral. So √In 1 In - dx X
Translate the lamina in Exercise 5 to the right five units and determine the resulting center of mass.Data from in Exercise 5Find the mass and center of mass of the lamina for each density.R: square with vertices (0, 0), (a, 0), (0, a),(a, a)(a) ρ = k (b) ρ = ky (c) ρ = kx
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. TT (cos 8 Jo r dr de
Use an iterated integral to find the area of the region bounded by the graphs of the equations.y = x, y = 2x + 2, x = 0, x = 4
Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. x = y = y 6 5 4 3 2 نرا (4u - v) (u = v) - (2, 2) R (6,3) (4,1) (0, 0)2 3 4 5 6 X
Sketch the region R and evaluate the iterated integral ∫R∫ ƒ(x,y) dA. [[°, Jo Jy-1 ex+y dx dy + 11-y ex+y dx dy
Sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. 2π √√√5 C5-r² Jo Jo Jo r dz dr de
Evaluate the integral. (π/2 sin³ x cos y dx
Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. x = // (u + v) y = 1/(u - v) y 2 1 (0, 1) (1, 2) R (3) 2 X
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. S.S. Jo 0 0 *sin 0 r² dr de
Use the result of Exercise 9 to make a conjecture about the change in the center of mass when a lamina of constant density is translated c units horizontally or d units vertically. Is the conjecture true when the density is not constant? Explain.Data from in Exercise 9Find the mass and center of
Use a computer algebra system to approximate the triple iterated integral. 2-(2y/3) 6-2y-3z ” Մ of ze-x²y² dx dz dy
Consider the functionFind the relationship between the positive constants a and k such that ƒ is a joint density function of the continuous randomvariables x and y. f(x, y) = [ke-(x+y)/a, 0, x ≥ 0, y ≥ 0 elsewhere.
Use an iterated integral to find the area of the region bounded by the graphs of the equations.x = y² + 1, x = 0, y = 0, y = 2
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 13 + x² - y², R = {(x, y): x² + y² ≤ 4}
Sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. 2T (π/2 Jo Jπ/6 Jo p² sin o dp do de
Evaluate the iterated integral. 2 [f² (x + y) dy dx JO
Sketch the region R whose area is given by the iterated integral.Then switch the order of integration and show that both ordersyield the same area. 5 If a 2 1 dx dy
Set upintegrals for both orders of integration. Use the more convenientorder to evaluate the integral over the region R.R: rectangle with vertices (0, 0), (0, 5), (3, 5), (3, 0) ffxy da ху dA JRJ
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. 2T (6 3r² sin 0 dr do
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = √x, y = 0, x = 1, ρ = ky
Find the volume of the solid generated by revolving the region in the first quadrant bounded by y = e-x2 about the y-axis. Use this result to find e-x² dx. 00
Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformations. x = (v - u) - - y = }(2v + u) y -ح -| 4 3 2R -1 تانیا - درا 20 | 2 3 نا
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = √x² + y², R = {(x, y): 0 ≤ ƒ(x, y) ≤ 1}
Evaluate the iterated integral by converting to polar coordinates. SS 0 9-12 (x² + y2)³/2 dy dx
Set up a triple integral for the volume of the solid.The solid in the first octant bounded by the coordinate planes and the plane z = 5 - x - y
Consider the surface ƒ(x, y) = x² + y² (see figure) and the surface area of f over each region R. Without integrating, order the surface areas from least to greatest. Explain.(a) R: rectangle with vertices (0, 0), (2, 0), (2, 2), (0, 2)(b) R: triangle with vertices (0, 0), (2, 0), (0, 2)(c) R =
Verify the given moment(s) of inertia and find x̅̅ and y̅̅. Assume that each lamina has a density of ρ = 1 gram per square centimeter. Ellipse b a 10=nab(a² + b²) X
To find the volume of each dome-shaped solid lying below the surface z = ƒ(x, y) and above the elliptical region R.(a)(b) f(x, y) = 16x² - y² x² y² + ≤ 1 16 9 R:
Use polar coordinates to set up and evaluate the double integral ∫R ∫ ƒ(x, y) dA. f(x, y) = 9x² - y² R: x² + y² ≤ 9, x ≥ 0, y ≥ 0
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations. 1 1 + y²⁹ x = 0, x = 2, y ≥ 0
List the six possible orders of integration for the triple integral over the solid regionQ = {(x, y, z): 0 ≤ x ≤ 2, x² ≤ y ≤ 4, 0 ≤ z ≤ 2 - x} Q, SSS .xyz dv. 2,
Evaluate the improper iterated integral. 8 x2 1 + y² dy dx
Set up a double integral to find the volume of the solid region bounded by the graphs of the equations. Do not evaluate the integral. X N Z=4-2x 2 y z=4-x² - y²
Consider the region R in the xy-plane bounded by the graph of the equation(a) Convert the equation to polar coordinates. Use a graphing utility to graph the equation.(b) Use a double integral to find the area of the region R.(c) Use a computer algebra system to determine the volume of the solid
Evaluate the improper iterated integral. ² ху dx dy
Sketch a graph of the region bounded by the graphs of the equations. Then use a double integral to find the area of the region.Inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ
List the six possible orders of integration for the triple integral over the solid regionQ = {(x, y, z): x² + y² ≤ 9,0 ≤ z ≤ 4} Q, SSS .xyz dv. 2,
Use spherical coordinates to find the volume of the solid.Solid inside x² + y² + z² = 9, outside z = √x² + y² and above the xy-plane
A company produces a spherical object of radius 25 centimeters. A hole of radius 4 centimeters is drilled through the center of the object.(a) Find the volume of the object.(b) Find the outer surface area of the object.
Set up a double integral to find the volume of the solid region bounded by the graphs of the equations. Do not evaluate the integral. -2 2 Z 4+ Z = x² + y² z = 2x 2 y
Find Ix, Iy, I0, , x̅̅ and y̅̅ for the lamina boundedby the graphs of the equations.y = 4 - x², y = 0, x > 0, ρ = kx-
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations.z = xy, x² + y2 = 1, first octant
A company builds a warehouse with dimensions 30 feet by 50 feet. The symmetrical shape and selected heights of the roof are shown in the figure.(a) Use the regression capabilities of a graphing utility to find a model of the form(b) Use the numerical integration capabilities of a graphing utility
Show that the surface area of the cone z = k√x² + y², k > 0, over the circular region x² + y² ≤ r² in the xy-plane is πr²√k² + 1 (see figure). N z=k√√x² + y² R: x² + y² ≤r² y
Use spherical coordinates to find the volume of the solid.Solid bounded above by x² + y² + z² = z and below by z = √x² + y²
Find the Jacobianfor the indicated change of variables. Ifthen the Jacobian of x, y, and z with respect to u, v, and w isx = 4u - v, y = 4v - w, z = u + w a(x, y, z) d(u, v, w)
Describe how to use the Jacobian to change variables in double integrals.
The figure shows the region of integration for the given integral. Rewrite the integral as an equivalent iterated integral in the five other orders. ° |z=9-x² (9-x² x20 y20 z≥0 X 3 dz dy dx Z 6 3 y = x 3 y
Find Ix, Iy, I0, , x̅̅ and y̅̅ for the lamina bounded by the graphs of the equations.y = x, y = x², ρ = kxy
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations.z = x² + y² + 3, z = 0, x² + y² = 1
Use spherical coordinates to find the volume of the solid.The solid between the spheresx² + y² + z² = a² and x² + y² + z² = b², b > a,and inside the cone z² = x² + y²
Set up a double integral to find the volume of the solid region bounded by the graphs of the equations. Do not evaluate the integral.z = sin² x, z = 0, 0 ≤ x ≤ π, 0 ≤ y ≤ 5
Find Ix, Iy, I0, , x̅̅ and y̅̅ for the lamina bounded by the graphs of the equations.y = x², y² = x, ρ = kx
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations.z = ln(x² + y²), z = 0, x² + y² ≥ 1, x² + y² ≤ 4
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = 2/x, y = 0, x = 1, x = 2, ρ = ky
From 1963 to 1986, the volume of the Great Salt Lake approximately tripled while its top surface area approximately doubled. Read the article "Relations between Surface Area and Volume in Lakes" by Daniel Cass and Gerald Wildenberg in The College Mathematics Journal. Then give examples of solids
Set up a triple integral for the volume of the solid.The solid bounded by z = 9 - x², z = 0, y = 0, and y = 2x
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. π/2 (3 STI 10 2 √9 r² r dr de -
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: sector of a circle in the first quadrant bounded byy = √25 - x², 3x - 4y =0, y = 0 J. fx da dA R.
Use a change of variables to find the volume of the solid region lying below the surface z = ƒ(x, y) and abovethe plane region R.ƒ(x, y) = 48xyR: region bounded by the square with vertices (1, 0), (0, 1),(1, 2), (2, 1)
Will the surface area of the graph of a function z = ƒx, y) over a region R increase when thegraph is shifted k units vertically? Why or why not?
Sketch the solid region whose volume is given by the iterated integral, and evaluate the iterated integral. 2T SIT 0 2 p² sin o dp do do
Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. [Lady 10 10 dy dx + x-9J 95 J3 J0 dy dx
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: rectangle with vertices (-π, 0), (π, 0), (π, π/2), (-π, π/2) JRJ sin x sin y dA
Evaluate the iterated integral. (x² - y²) dy dx -1J-2
Convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. -2. √√4-x² √4-x²√x² + y² x dz dy dx
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. π/4 (4 [™ 0 0 r2 sin 0 cos 0 dr de
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = x², y = 0, x = 2, ρ = kxy
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = xy, R = {(x, y): x² + y² ≤ 16}
Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. 48. dy dx Jo J2x
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: trapezoid bounded by y = x, y = 2x, x = 1, x = 2 SS= y x² + y² +y dA
The angle between a plane P and the xy-plane is θ, where 0 ≤ θ < π/2. The projection of a rectangular region in P onto the xy-plane is a rectangle whose sides have lengths Δx and Δy, as shown in the figure. Prove that the area of the rectangular region in P is sec θ Δx Δy. Area: sec θ
Evaluate the iterated integral. KS (x² - 2y²) dx dy
Convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. CLAS 0 0 √4-x² √16-x²-y² √x² + y² dz dy dx
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = 4/x, y = 0, x = 1, x = 4, ρ = kx²
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. y || 1 1 + x²² y = 0₁ x = -1, x = 1, p = k 0, 0,
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = √a² - x² - y²R = {(x, y): x² + y² ≤ b²,0 < b < a}
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. π/2 (3 Jo re-r² dr de
Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area. (9-y² -3 JO dx dy
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: triangle bounded by y = 4 - x, y = 0, x = 0 R. xey dA
Set up a triple integral for the volume of the solid.The solid bounded by z = 6 - x² - y² and z = 0
Use the indicated change of variables to evaluate the double integral. JRJ 40 4(x² + y²) dA X = 1/(u + v) y = (u - v)
Evaluate the iterated integral. SS.Cx -1J1 (x + y²) dx dy
Setup integrals for both orders of integration. Use the moreconvenient order to evaluate the integral over the region R. √√√4x R. R: rectangle with vertices (0, 0), (0, 4), (2, 4), (2, 0) 4xy dA
Convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. a²-x² IIZI /a²-x² ra+ √a²-x²-y² -a x dz dy dx
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = √a² - x² − y ²R = {(x, y): x² + y² ≤ a²}
Evaluate the iterated integral. π/2 Sox y cos x dy dx
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. m/2 *1+sin 6 ¨ 0 Or dr de
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: region bounded by y = 4 - x², y = 4 - x S.S - 2y dA
Set up a triple integral for the volume of the solid.The solid bounded by z = √16 - x² - y² and z = 0
Convert the integral from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simplest iterated integral. √9-x² [*T 0 10 √9-x²-y² /x² + y² + z² dz dy dx
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = ex, y = 0, x = 0, x = 1, ρ = k
Consider a circular lawn with a radius of 10 feet, as shown in the figure. Assume that a sprinkler distributes water in a radial fashion according to the formula(measured in cubic feet of water per hour per square foot of lawn), where r is the distance in feet from the sprinkler. Find the amount of
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: region bounded by y = 0, y = √x, x = 4 R. y 1 + x² dA
Use the indicated change of variables to evaluate the double integral. 60xy dA JRJ x = 1/2 (u + v) y = -(u - v)

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