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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

Find the area of the surface.The portion of the plane z = 24 - 3x - 2y in the first octant
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R. Sfax² R: region bounded by y = 0, y = √√x, x = 1 6x² dA
Set up a triple integral for the volume of the solid.The solid that is the common interior below the sphere x² + y² + z² = 80 and above the paraboloid z = 1/2(x² + y²)
Evaluate the iterated integral. In 4 In 3 0 ex+y dy dx
Use the indicated change of variables to evaluate the double integral. √√√x(x - y(x - y) dA R x = u + v y = u 6 4 2 -2 (3, 3) (7,3) 27 R (0, 0) (4,0) 6 8 X
Evaluate the double integral ∫R ∫ ƒ(r, θ) dA, and sketch the region R. *π/2 1-cos 8 ["²" 0 0 (sin 0)r dr de
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = e-x, y = 0, x = 0, x = 1, ρ = ky²
Sketch the solid whose volume is given by the sum of the iterated integralsThen write the volume as a single iterated integral in the order dy, dz, dx. Jz/2Jz/2 r6 (12-2)/2 6-y Toz Jz/2 dx dy dz +
Find the area of the surface.The portion of the paraboloid z = 16 - x² - y² in the first octant
Evaluate the iterated integral by converting to polar coordinates. √₁²-y² JJ 0 y dx dy
Use a double integral to find the volume of the indicated solid. z = 5-x X 3. 5 2 y 0≤x≤3 0≤y≤2
Use a triple integral to find the volume of the solid shown in the figure. X 4 |x=4-y²| Z=X Z=0 Y Z 4- 3
Set up a triple integral for the volume of the solid.The solid bounded above by the cylinder z = 4 - x² and below by the paraboloid z = x² + 3y²
Use the indicated change of variables to evaluate the double integral. Sof 40 R x = 1/(u + v) y = (u - v) 4(x + y)ex-y dA (-1, 1) R y (1, 1) + (0,0) i at X
Evaluate the iterated integral. C.S. *sin x 0 0 (1 + cos x) dy dx
Use cylindrical coordinates to find the volume of the solid.Solid inside both x² + y² + z² = a²and (x - a/2)² + y² = (a/2)²
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = 4 - x², y = 0, ρ = ky
The figure shows a solid bounded below by the plane z = 2 and above by the sphere x² + y² + z² = 8.(a) Find the volume of the solid using cylindrical coordinates.(b) Find the volume of the solid
Use a double integral to find the volume of the indicated solid. Z=4 X Z |x=2 y = x 2 -y
Find the area of the surface.The portion of the sphere x² + y² + z² = 25 inside the cylinder x² + y² = 9
Evaluate the iterated integral. -4 S.S. 1 J1 2ye* dy dx
Evaluate the iterated integral by converting to polar coordinates. √a²-x² J.J JO JO x dy dx
Use a triple integral to find the volume of the solid shown in the figure. Z 00 8 -6 4 2 z = 2xy X 3 0≤x≤2 0≤y≤2
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the region R.R: semicircle bounded by y = √4 - x², y = 0 JRJ (x² + y²) dA (x²
Use cylindrical coordinates to find the volume of the solid.Solid inside x² + y² + z² = 16 and outside z = √x² + y²
Find the area of the surface.The portion of the cone z = 2√x² + y² inside the cylinder x² + y² = 4
Use the indicated change of variables to evaluate the double integral. x = 3 2 y -xy/2 dA √ V y = R 1 X y=2x 4 y= X نا UV 3 y= X JAKO X
Use a double integral to find the volume of the indicated solid. X 4 Z |z=4-x² - y²| 2 -1≤x≤1 -1≤y≤1
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.x = 9 - y², x = 0, ρ = kx
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. y = sin π.Χ. , y = 0, x = 0, x = L, p = k L
Use the indicated change of variables to evaluate the double integral. SS» R X 3 2 y y sin xy dA u y = v xy=1] |y=4 R xy=4 12 نیا y = 1 4
Use a double integral to find the volume of the indicated solid. X 4 3 2 1. 1 Z 0≤x≤4 0≤y≤2 Z= 2 y
Evaluate the iterated integral. 0 0 √1-x² dy dx
Evaluate the iterated integral by converting to polar coordinates. 2 √√4-1² LG -2 -2 JO (x² + y²) dy dx
Use cylindrical coordinates to find the volume of the solid.Solid bounded above by z = 2x and below by z = 2x² + 2y²
Use a triple integral to find the volume of the solid shown in the figure. X Z a y ₂D = z2+za+zx|
Write a double integral that represents the surface area of z = ƒ(x, y) over the region R. Use a computer algebra systemto evaluate the double integral.ƒ(x, y) = 2y + x², R: triangle with vertices
Use a double integral to find the volume of the indicated solid. X 2 Z x+y+z=2 2
Evaluate the iterated integral by converting to polar coordinates. √x-x² √x-x² (x² + y²) dy dx
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. y = cos π.Χ. L L x = = ₁ p = ky 2² , y = 0, x = 0, x
Use a double integral to find the volume of the indicated solid. X Z 6 z=6-2y 2 0≤x≤4 0≤y≤2
Evaluate the iterated integral. LS² -4J0 √64 - x³ dy dx
Use cylindrical coordinates to find the volume of the solid.Solid bounded above by z = 2 - x² - y² and below by z = x² + y²
Use a triple integral to find the volume of the solid shown in the figure. X 36 12 |z=36-x² - y2 12 Z=0 y
Write a double integral that represents the surface area of z = ƒ(x, y) over the region R. Use a computer algebra system to evaluate the double integral.ƒ(x, y) = 2x + y2, R: triangle with vertices
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. y = √√a²-x², 0 ≤ y ≤ x, p = k
Use a double integral to find the volume of the indicated solid. X Z 41 3 21 y=x |z=4-x-y 2 y = 2
Evaluate the iterated integral. 3y [² S²( ³ + x² + 1/3²) dx dy 3 -1J0
Use cylindrical coordinates to find the volume of the solid.Solid bounded by the graphs of the sphere r² + z² = a² and the cylinder r = a cos θ
Write a double integral that represents the surface area of z = ƒ(x, y) over the region R. Use a computer algebra system to evaluate the double integral.ƒ(x, y) = 9 - x² - y², R = {(x, y): 0 ≤
Find the average value of ƒ(x, y) over the plane region R.ƒ(x) = 16 - x² - y²R: rectangle with vertices (2, 2), (-2, 2), (-2, -2), (2, -2)
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = 4 - x², y = 4 - x², first octant
Use a change of variables to find the volume of the solid region lying below the surface z = ƒ(x, y) and above the plane region R.ƒ(x, y) = (3x + 2y)² √2y - xR: region bounded by the
Use cylindrical coordinates to find the volume of the solid. Solid inside the sphere x² + y² + z² = 4 and above the upper nappe of the cone z² = x² + y²
Write a double integral that represents the surface area of z = ƒ(x, y) over the region R. Use a computer algebra system to evaluate the double integral.ƒ(x, y) = x² + y², R = {(x, y): 0 ≤
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.x² + y² = a², x ≥ 0, y ≥ 0, ρ = k(x² + y²)
Use a double integral to find the volume of the indicated solid. X 2x + 3y + 4z = 12 31
Evaluate the iterated integral by converting to polar coordinates. √√8-1² 0 Jy x² + y² dx dy 2
Evaluate the iterated integral. 2y [f²³ 10 y (10 + 2x² + 2y²) dx dy
The temperature in degrees Celsius on the surface of a metal plate iswhere x and y are measured in centimeters. Estimate the average temperature when x varies between 0 and 3 centimeters and y varies
Find the average value of ƒ(x, y) over the plane region R.ƒ(x) = 2x² + y²R: square with vertices (0, 0), (3, 0), (3, 3), (0, 3)
Use cylindrical coordinates to find the mass of the solid Q of density ρ. Q = {(x, y, z): 0 ≤ z ≤ 9 - x - 2y, x² + y² ≤ 4} p(x, y, z) = k√x² + y²
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = 9 - x³, y = -x² + 2, y = 0, z = 0, x ≥ 0
Use a double integral to find the volume of the indicated solid. X N y = x z=1-xy [y=1]
Use a change of variables to find the volume of the solid region lying below the surface z = ƒ(x, y) and above the plane region R.ƒ(x, y) = (x + y)ex-yR: region bounded by the square with vertices
Evaluate the iterated integral. Jo 1-y² (x + y) dx dy
Use cylindrical coordinates to find the mass of the solid Q of density ρ. Q = {(x, y, z): 0 ≤ z ≤ 12e-(x² + y²), x² + y² ≤ 4, x ≥ 0, y ≥ 0} p(x, y, z) = k
Write a double integral that represents the surface area of z = ƒ(x, y) over the region R. Use a computer algebra system to evaluate the double integral.ƒ(x, y) = 4 - x² - y²R = {(x, y): 0 ≤ x
Evaluate the iterated integral by converting to polar coordinates. SS lo Jo 2x-x² xy dy dx
Use a computer algebra system to find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = e-x, y = 0, x = 0, x = 2, ρ = kxy
Use a double integral to find the volume of the indicated solid. X 4 3 2+ y = x z=4-y² y = 2 -y
A firm’s profit P from marketing two soft drinks iswhere x and y represent the numbers of units of the two soft drinks. Estimate the average weekly profit when x varies between 40 and 50 units and
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = 2 - y, z = 4 - y², x = 0, x = 3, y = 0
Evaluate the iterated integral. (2y-y² J3y²-6y 3y dx dy
Use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure.Find the volume of the cone. X h |z=h(1-7) ro
Use a change of variables to find the volume of the solid region lying below the surface z = ƒ(x, y) and above the plane region R.ƒ(x, y) = (x + y)² sin²(x - y)R: region bounded by the square
Write a double integral that represents the surface area of z = ƒ(x, y) over the region R. Use a computer algebra system to evaluate the double integral.ƒ(x, y) = 2/3x³/2 + cos xR = {(x, y): 0 ≤
Evaluate the iterated integral by converting to polar coordinates. 오 10 Jo | 4y-y2 x² dx dy
Use a computer algebra system to find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. r = 2 cos 30, 6 TT 6' ≤0 ≤- p = k
Use a computer algebra system to find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = ln x, y = 0, x = 1, x = e, ρ =k/x
Use a double integral to find the volume of the indicated solid. X Improper integral Z Z= 1 (x + 1)²(y + 1)² 0≤x
Evaluate the iterated integral. √4-y SS 0 2 4- y² dx dy
Evaluate the iterated integral by converting to polar coordinates. S.S." √1-x²2² cos(x² + y²) dy dx
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 dz dx. xp Ap zp -1J0 JJJ
Evaluate the iterated integral by converting to polar coordinates. ch SS 10 x² + y² dy dx
Use a triple integral to find the volume of the solid bounded by the graphs of the equations.z = x, y = x + 2, y = x², first octant
Use a change of variables to find the volume of the solid region lying below the surface z = ƒ(x, y) and above the plane region R.ƒ(x, y) = √(x - y)(x + 4y)R: region bounded by the parallelogram
Set upa double integral that gives the area of the surface on the graphof ƒ over the region R.ƒ(x, y) = exy, R = {(x, y): 0 ≤ x ≤ 4,0 ≤ y ≤ 10}
Use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure.Find the centroid of the cone. X h |z=h(1-7) ro
Use a double integral to find the volume of the indicated solid. 2. Improper integral |z=e=(x+y)/2 0≤x
Evaluate the iterated integral by converting to polar coordinates. √16-y² ST" 10 (x² + y²) dx dy
Use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure.Find the center of mass of the cone, assuming that its density at any point is proportional to the
Evaluate the iterated integral. Jo 4 + y2 dx dy x² +
Use a change of variables to find the volume of the solid region lying below the surface z = ƒ(x, y) and above the plane region R.ƒ(x, y) = (3x + 2y) (2y - x)³/2R: region bounded by the
Evaluate the iterated integral by converting to polar coordinates. √4-x² Jo sin x² + y² dy dx
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 dx dz dy. . | -1.Jy- *1-x ս dz dx dy
Set up a double integral that gives the area of the surface on the graph of ƒ over the region R.ƒ(x, y) = x² - 3xy - y²R = {(x, y): 0 ≤ x ≤ 4,0 ≤ y ≤ x}
Use a computer algebra system to find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density. r =1 + cos θ, ρ = k
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. (These regions are common shapes used in
Evaluate the iterated integral. */2 *sin 9 « Or dr de

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