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mathematics
calculus 10th edition

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

Determine the location of the horizontal axis ya at which a vertical gate in a dam is to be hinged so that there is no moment causing rotation under the indicated loading (see figure). The model for ya iswhere Y̅ is the y-coordinate of the centroid of the gate, IY̅ is the moment of inertia of the
Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations.z = ln(1 + x + y), z = 0, y = 0, x = 0, x = 4 - √y
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 8 + 4x - 5yR = {(x, y): x² + y² ≤ 1}
Use a double integral to find the area of the shaded region. R|N I 2 r = 1 + cos 0 1 0
Use an iterated integral to find the area of the region bounded by the graphs of the equations.y = 4 - x², y = x + 2
Use a double integral to find the area of the shaded region. RINHH ✈0 234 r=2+ sin 0
Give the equations for conversion from rectangular to cylindrical coordinates and vice versa.
The center of mass of a solid of constant density is shown in the figure. Make a conjecture about how the center of mass (x̅, y̅, z̅ ) will change for the nonconstant density ρ(x, y, z). Explain.ρ(x, y, z) = kz (2,0,3) 73२ X 3 نرا Z 21
Let ƒ be a continuous function such that 0 ≤ ƒ(x, y) ≤ 1 over a region R of area 1. Prove that 0 ≤ ∫R∫ ƒ(x, y)da ≤ 1.
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 9 - y²R: triangle bounded by the graphs of the equations y = x,y = -x, and y = 3
Use a double integral to find the area of the shaded region. 几 2 | r = 2 sin 30 12 -0
The center of mass of a solid of constant density is shown in the figure. Make a conjecture about how the center of mass (x̅, y̅, z̅ ) will change for the nonconstant density ρ(x, y, z). Explain.ρ(x, y, z) = k(y + 2) (2,0,3) 73२ X 3 نرا Z 21
Sketch the region R of integration and switch the order of integration. [ f* f(x, y) dx dy
Determine the location of the horizontal axis ya at which a vertical gate in a dam is to be hinged so that there is no moment causing rotation under the indicated loading (see figure). The model for ya iswhere Y̅ is the y-coordinate of the centroid of the gate, IY̅ is the moment of inertia of the
Give the iterated form of the triple integral in cylindrical form. SSS f(x, y, z) dv dV Q
Use an iterated integral to find the area of the region bounded by the graphs of the equations.y = x, y = 2x, x = 2
Sketch the region of integration. Then evaluate the iterated integral, switching the order of integration if necessary. 1/2 Jo Jy/2 ex dx dy
Give the equations for conversion from rectangular to spherical coordinates and vice versa.
Find the volume of the solid in the first octant bounded by the coordinate planes and the plane (x/a) + (y/b) + (z/c) = 1, where a > 0, b > 0, and c > 0.
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 4 - x²R: triangle bounded by the graphs of the equations y = x,y = -x, and y = 2
The center of mass of a solid of constant density is shown in the figure. Make a conjecture about how the center of mass (x̅, y̅, z̅ ) will change for the nonconstant density ρ(x, y, z). Explain.ρ(x, y, z) = kxz²(y + 2)² (2,0,3) 73२ X 3 نرا Z 21
Use a double integral to find the area of the shaded region. RIN r = 3 cos 20 -0
Sketch the region R of integration and switch the order of integration. Jo f(x, y) dx dy
Sketch the region R of integration and switch the order of integration. S LE -2 JO √4-x² f(x, y) dy dx
Give the iterated form of the triple integral in spherical form. Ꮽ . [ f (x, y, z) dᏙ [ dV
Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) Z || h √x² + y², z = h
Give the formulas for finding the moments and center of mass of a variable density planar lamina.
Sketch the region of integration. Then evaluate the iterated integral, switching the order of integration if necessary. In 10 10. 1 for To my d 10 ex dy dx
Give the formulas for finding the moments of inertia about the x- and y-axes for a variable density planar lamina.
The roof over the stage of an open air theater at a theme park is modeled bywhere the stage is a semicircle bounded by the graphs of(a) Use a computer algebra system to graph the surface.(b) Use a computer algebra system to approximate the number of square feet of roofing required to cover the
Evaluate the triple iterated integral. - SSS 0 0 4 (2x + y + 4z) dy dz dx
Sketch the region of integration. Then evaluate the iterated integral, switching the order of integration if necessary. √4-x² LE -2J -√4-x² 4- y² dy dx
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 = 2 cos θ and outside the circle r = 1
Sketch the region R of integration and switch the order of integration. So So 0 (4-x² f(x, y) dy dx
Sketch the region of integration. Then evaluate the iterated integral, switching the order of integration if necessary. 1 Jy/3 0 1 + x4 dx dy
Evaluate the triple iterated integral. xy IIT 0 0 y dz dx dy
Describe the surface whose equation is a coordinate equal to a constant for each of the coordinates in (a) The cylindrical coordinate system(b) The spherical coordinate system
Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) y = √√√√9-x², z = y, z = 0
The solid is bounded below by the upper nappe of a cone and above by a sphere (see figure). Would it be easier to use cylindrical coordinates or spherical coordinates to find the volume of the solid? Explain. Upper nappe of cone: |z² = x² + y² y Sphere: x² + y² + z² = 4
Sketch the region R of integration and switch the order of integration. 10 In y I.T."* f(x, y) dx dy
In your own words, describe what the radius of gyration measures.
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 cardioid r = 2 + 2 cos θ and outside the circler = 1
Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) 7 = 16 – x2 - y2 = 0
Prove the following 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. If r is the distance between the centroid of R and the line, then the volume V of the solid of revolution formed by revolving R about the line
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 cose θ and outside the cardioid r = 1 + cos θ
Sketch the region of integration. Then evaluate the iterated integral, switching the order of integration if necessary. SS Jo Jo arccos y sin x 1 + sin² x dx dy
Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) Z = 1 2 y² + 1' z = 0, x = -2, x = 2, y = 0, y = 1
Evaluate the triple iterated integral. JJJ 0 0 (x² + y² + z²) dx dy dz
Sketch the region of integration. Then evaluate the iterated integral, switching the order of integration if necessary. So J(1/2)x² √y cos y dy dx
Sketch the region R of integration and switch the order of integration. L. S 10 f(x, y) dy dx
Evaluate the triple iterated integral. S.J 5 z sin x dy dx dz Jπ/2J2
Find the volume of the region of points (x, y, z) such that (x² + y² + z² + 8)² ≤ 36(x² + y²).
Sketch the region R of integration and switch the order of integration. -1√x² f(x, y) dy dx
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 cardioid r = 1 + cos θ and outside the circler = 3 cos θ
Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) X 20 cm Z -12 cm. 15 cm y
Find the centroid of the solid region bounded by the graphs of the equations or described by the figure. Use a computer algebra system to evaluate the triple integrals. (Assume uniform density and find the center of mass.) X (0, 0, 4) (5, 0, 0) Z (0,3,0)
Use a computer algebra system to approximate the iterated integral. √1-x² √1-x²-y² IM √1-x²-√√√1-x² - y² (x² + y2) dz dy dx
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 rose curve r = 4 sin 3θ and outside the circle r = 2
Sketch the region R of integration and switch the order of integration. π/2 cos x J-T/2 Jo f(x, y) dy dx
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. SS² 0 dy dx
Find the average value ofƒ(x, y) over the plane region R.ƒ(x, y) = xR: rectangle with vertices (0, 0), (4, 0), (4, 2), (0, 2)
Find Ix, Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals.(a) ρ = k (b) ρ = kxyz a X a a y
Use a computer algebra system to approximate the iterated integral. /4-x²√√4-x²-y² ff 10 0 xyz dz dy dx
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 = 2 and outside the cardioid r = 22 cos θ
Find the average value of ƒ(x, y) over the plane region R.ƒ(x, y) = 2xyR: rectangle with vertices (0, 0), (5, 0), (5, 3), (0, 3)
Find Ix, Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals.(a) ρ(x, y, z) = k (b) ρ(x, y, z) = k(x² + y²) N ND y GIN GIN X
Describe the partition of the region R of integration in the xy-plane when polar coordinates are used to evaluate a double integral.
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. 4 J2 dx dy
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. 1-y² -√1-y² dx dy
Find the average value of ƒ(x, y) over the plane region R.ƒ(x, y) = x² + y²R: square with vertices (0, 0), (2, 0), (2, 2), (0, 2)
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = xy, z = 0, 0 ≤ x ≤ 3, 0 ≤ y ≤ 4
Explain how to change from rectangular coordinates to polar coordinates in a double integral.
Find the average value of ƒ(x, y) over the plane region R.ƒ(x, y) = 1/x + yR: triangle with vertices (0, 0), (1, 0), (1, 1)
Find Ix, Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals.(a) ρ(x, y, z) = k (b) ρ = ky X 4 N |Z=4-x| 4
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = 8 - x - y, z = 0, y = x, y = 3, x = 0
In your own words, describe r-simple regions and θ-simple regions.
Sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration.Rewrite using the order dz dy dx. ISI™ √1-x² dz dx dy
Find the average value of ƒ(x, y) over the plane region R.ƒ(x, y) = ex+yR: triangle with vertices (0, 0), (0, 1), (1, 1)
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. L -2 √4-x² √4-x² dy dx
Let R be the region bounded by the circle x² + y² = 9.(a) Set up the integral (b) Convert the integral in part (a) to polar coordinates.(c) Which integral would you choose to evaluate? Why? S. ff(x, y) da.
Find Ix, Iy, and Iz for the solid of given density. Use a computer algebra system to evaluate the triple integrals.(a) ρ = kz (b) ρ = k(4 - z) X + N |z=4-y²| 2
Verify the moments of inertia for the solid of uniform density. Use a computer algebra system to evaluate the triple integrals. X L₂ = 1/2m(3a² + L²) 1₁ = 1/ma² 1₂ = 1/2m(3a² + L²) X L N 42 2 y
Sketch the solid whose volume is given by the iterated integral and rewrite the integral using the indicated order of integration.Rewrite using the order dy dx dz. xp Ap zp 0/ do A-x-9J x-9 9 О
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. S.S. 0 Jo 4 (4-x + [*[*** 2 dy dx + dy dx
The population density of a city is approximated by the modelfor the regionwhere and are measured in miles. Integrate the density function over the indicated circular region to approximate the population of the city. f(x, y) = 4000e-0.01(x² + y²)
Find the average value of ƒ(x, y) over the plane region R.ƒ(x, y) = sin(x + y)R: rectangle with vertices (0, 0), (π, 0), (π, π), (0, π)
The Cobb-Douglas production function for an automobile manufacturer iswhere x is the number of units of labor and y is the number of units of capital. Estimate the average production level when the number of units of labor x varies between 200 and 250 and the number of units of capital y varies
Find the mass and the indicated coordinates of the center of mass of the solid region Q of density ρ bounded by the graphs of the equations. Find using p(x, y, z) = k. Q: x + y + z = 10, x = 0, y = 0, z = 0
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. S (x/2 0 0 dy dx + x-9J 95 4 J0 dy dx
Verify the moments of inertia for the solid of uniform density. Use a computer algebra system to evaluate the triple integrals. 1₂ = 1/2m(a² + b²) x 1₁ = 1/2m(b² + c²) 1₂ = 1/2m(a² + c²) X 0/2 6/2 N BIN
The temperature in degrees Celsius on the surface of a metal plate isT(x, y) = 20 - 4x² - y² where x and y are measured in centimeters. Estimate the average temperature when x varies between 0 and 2 centimeters and y varies between 0 and 4 centimeters.
Find the mass and the indicated coordinates of the center of mass of the solid region Q of density ρ bounded by the graphs of the equations. Find y using p(x, y, z) = kx. Q: z = 5-y, z = 0, y = 0, x = 0, x = 5
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. 2 10 Jx/2 dy dx
Set up a tripleintegral that gives the moment of inertia about the z-axis of thesolid region Q of density ρ. Q = {(x, y, z): -1 ≤ x ≤ 1, − 1 ≤ y ≤ 1,0 ≤ z ≤1 − x} p = √√√x² + y² + 2²
Determine the diameter of a hole that is drilled vertically through the center of the solid bounded by the graphs of the equations z = 25e-(x²+y²)/4, z = 0, and x² + y² = 16when one-tenth of the volume of the solid is removed.
Evaluate the iterated integral. +J ε/"] JJJ 0 Jo Jo r cos 0 dr do dz
State the definition of a double integral. When the integrand is a nonnegative function over the region of integration, give the geometric interpretation of a double integral.
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 3. I.T. √x dy dx
Using the description of the solid region, set up the integral for (a) The mass(b) The center of mass(c) The moment of inertia about the z-axis.The solid bounded by z = 4 - x² - y² and z = 0 with density function ρ = kz
Use a computer algebra system to approximate the iterated integral. TT/2 (5 So TT/4 Jo r√1+r³ sindr de

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