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mathematics
calculus 10th edition

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

Verify the given moment(s) of inertia and find x̅̅ and y̅̅. Assume that each lamina has a density of ρ = 1 gramper square centimeter. (These regions are common shapes usedin engineering.) Rectangle y h b L = { bh³ = X
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 + yR: region bounded by the triangle with vertices (0, 0), (a, 0),(0, a), where a > 0
Set up a double integral that gives the area of the surface on the graph of ƒ over the region R.ƒ(x, y) = e-x sin y, R = {(x, y): x² + y² ≤ 4}
The region R is transformed into a simpler region S (see figure). Which substitution can be used to make the transformation?(a) u = 3y - x, v = y - x (b) u = y - x, v = 3y - x 5 y (6,4) (3, 3) 这 R (4,2) 4 3- 2 1 (1, 1) 111 1 2 3 4 5 6 X (-2, 6) (-2,2) S 5+ 3 V 111 -5-4-3-2-1 -
Use polar coordinates to set up and evaluate the double integral∫R ∫ ƒ(x, y) dA. f(x, y) = x + y R: x² + y² ≤ 4, x ≥ 0, y ≥ 0
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations.z = xy², x² + y² = 9, first octant
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. √1-y² CIC 10 dz. dx dy
Use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure.Assume that the density of the cone isand find the moment of inertia about the z-axis. X h |z=h(1-7) ro
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations.z = xy, z = 0, y = x, x = 1, first octant
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 engineering.) Semicircle y a X 10 ==nat
Evaluate the iterated integral. π/4 cos 0 10 3r² sin 0 dr do
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations.x² + z² = 1, y² + z² = 1, first octant
Use polar coordinates to set up and evaluate the double integral ∫R ∫ ƒ(x, y) dA. f(x, y) = e-(x² + y²)/2 R: x² + y2 ≤ 25, x ≥ 0
Use a double integral to find the area of the shaded region. RIN- 2 r=2 sin 20 8 2 + -0
Answer each question about the surface area S on a surface given by a positive function z = ƒ(x, y) over a region R in the xy-plane. Explain each answer.(a) Is it possible for S to equal the area of R?(b) Can S be greater than the area of R?(c) Can S be less than the area of R?
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 dy dz. /y²-4x² SEC J0 J2x JO dz dy dx
Use cylindrical coordinates to verify the given formula for the moment of inertia of the solid of uniform density. Cylindrical shell: I = m(a² + b²) 0
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 engineering.) Quarter circle y a lo=gnat X
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations.y = 4 - x², z = 4 - x², first octant -
Evaluate the improper iterated integral. So roo (1/x 0 y dy dx
Use polar coordinates to set up and evaluate the double integral ∫R ∫ ƒ(x, y) dA. f(x, y) = arctan X R: x² + y²1, x² + y² ≤ 4,0 ≤ y ≤ x
List the six possible orders of integration for the triple integral over the solid regionQ = {(x, y, z): 0 ≤ x ≤ 1,0 ≤ y ≤ x, 0 ≤ z ≤ 3} Q, SSS .xyz dv. 2,
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations.z = x + y, x² + y² = 4, first octant
Use cylindrical coordinates to verify the given formula for the moment of inertia of the solid of uniform density.Use a computer algebra system to evaluate the triple integral. Right circular r = 2a sin 0, cylinder: 1₂ = 2/ma² 0≤z ≤h
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 circle r = 3
Evaluate the iterated integral. π/2 (2 cos 0 8 0 r dr do
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. 4 (4-x)/2 (12-3x-6y)/4 ՞ 0 dz dy dx
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. f(x, y) = ху 1 + x²y² R: region bounded by the graphs of xy = 1, xy = 4, x = 1, x = 4
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 distance between the point and the base. Use a computer algebra system to evaluate the triple integral.
Set up a double integral that gives the area of the surface on the graph of ƒ over the region R. = {(x, y): x²2 f(x, y) = cos(x² + y²), R=(x, y): x² + y² 2
Combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting iterated integral. 2²-8/12/20 [²[* √x + y dy dx + [2x² [² x² J2 0 x² + y² dy dx
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations. z = √25x² - y², z = 0, x² + y² = 16
Combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting iterated integral. 5√2/2 5 √25-x² [12/²3/[ 1 xy dy dx + √2/2 √² 0 xy 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 dz dx dy. SS. 0 9-x² (6-x-y dz dy dx
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 engineering.) Right triangle y h b 4 = 1/2 bh³ 1₂ = 1/2b³h X
Evaluate the iterated integral. π/4 √√3 cos 0 [S انا r dr do
The substitutions u = 2x - y and v = x + y make the region R (see figure) into a simpler region S in the uv-plane. Determine the total number of sides of S that are parallel to either the u-axis or the v-axis. 8 6 4 2 y (2,7) R 2 (0, 0) 4 (6, 3) + 6 8 X
Use cylindrical coordinates to find the indicated characteristic of the cone shown in the figure.Assume that the cone has uniform density and show that the moment of inertia about the z-axis is X h |z=h(1-7) ro
Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations.z = 0, z = x², x = 0, x = 2, y = 0, y = 4
Use a double integral to find the area of the shaded region. RIN- r = 2 + cos 0 12 4 ✈0
State the double integral definition of the area of a surface S given by z = ƒ(x, y) over the regionR in the xy-plane.
Combine the sum of the two iterated integrals into a single iterated integral by converting to polar coordinates. Evaluate the resulting iterated integral. (8/√13 (3x/2 10 10 xy dy dx + J8/√13 Jo /16-x² xy dy dx
List the six possible orders of integration for the triple integral over the solid regionQ = {(x, y, z): 0 ≤ x ≤ 1, y ≤ 1 - x²,0 ≤ z ≤ 6} Q, SSS .xyz dv. 2,
Evaluate the improper iterated integral. [.. 0 0 xye-(x²+y²) dx dy
Find the Jacobianfor the indicated change of variables. Ifthen the Jacobian of x, y, and z with respect to u, v, and w isx = u(1 - v), y = uv(1 - w), z = uvw a(x, y, z) d(u, v, w)
Find the surface area of the solid of intersection x² + z² = 1 and y2 + z² = 1 (see figure). of the cylinders -3 X 3 21 |y² + z² = 1 3 x²+z² = 1
Use an iterated integral to find the area of the region. 00 У 8 6- 4 2 2 4 6 (8,3) 8 X
Use spherical coordinates to find the volume of the solid.The torus given by ρ = 4 sin Ø (Use a computer algebrasystem to evaluate the triple integral.)
Use an iterated integral to find the area of the region. 3 نا 2 1 y | y = 4 – x2 - 1 + 3 4
The figure shows the region of integration for the given integral. Rewrite the integral as an equivalent iterated integral in the five other orders. x20 y 20 Z≥0 0 z x -y²f1-y 0 dz dx dy x = 1 - y² z = 1-y
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 = x² + y², x² + y² = 4, z = 0
Use an iterated integral to find the area of the region. نرا 3 2 y (1, 3) (2,3) (1, 1) + (2, 1) + 12 3 + r
Find Ix, Iy, I0, , x̅̅ and y̅̅ for the lamina bounded by the graphs of the equations.y = √x, y = 0, x = 4, 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, z = 0, x2 + y2 = 25
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = x³, y = 0, x = 2, ρ = kx
Find the Jacobianfor the indicated change of variables. Ifthen the Jacobian of x, y, and z with respect to u, v, and w is a(x, y, z) d(u, v, w)
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: 2x + 3y + 6z = 12, x = 0, y = 0, z = 0
Use spherical coordinates tofind the mass of the sphere x² + y² + z² = a² with the givendensity.The density at any point is proportional to the distance between the point and the origin.
Use an iterated integral to find the area of the region. 5 4 3 2 نرا 1 y y= 1 √x-1 2≤x≤5 1 2 3 4 5 X
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 = x² + 2y², z = 4y
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations.Inside the hemisphere z = √16 - x² - y² and inside the cylinder x² + y² - 4x = 0
Find the Jacobianfor the indicated change of variables. Ifthen the Jacobian of x, y, and z with respect to u, v, and w isx = u - v + w, y = 2uv, z = u + v + w a(x, y, z) d(u, v, w)
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = 2x, y = 2x³, x ≥ 0, y ≥ 0, ρ = kxy
Set up the double integral required to find themoment of inertia I, about the given line, of the lamina boundedby the graphs of the equations. Use a computer algebra systemto evaluate the double integral.x² + y² = b², ρ = k, line: x = a (a > b)
Find the Jacobianfor the indicated change of variables. Ifthen the Jacobian of x, y, and z with respect to u, v, and w isx = ρ sin Ø cos θ, y = ρ sin Ø sin θ, z = ρ cos Ø a(x, y, z) d(u, v, w)
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) = ky. Q: 3x + 3y + 5z = 15, x = 0, y = 0, z = 0
Use spherical coordinates to find the mass of the sphere x² + y² + z² = a² with the given density.The density at any point is proportional to the distance of the point from the z-axis.
Find a such that the volume inside the hemisphereand outside the cylinderis one-half the volume of the hemisphere. z = √16x² - 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 = x² + y², z = 18 - x² - y²
Use a double integral in polar coordinates to find the volume of the solid bounded by the graphs of the equations.Inside the hemisphere z = √16 - x² - y² and outside the cylinder x² + y² = 1
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 z using p(x, y, z) = kx. Q: z = 4x, z = 0, y = 0, y = 4, x = 0
Find the mass and center of mass of the lamina bounded by the graphs of the equations for the given density.y = 6 - x, y = 0, x = 0, ρ = kx²
Set up the double integral required to find the moment of inertia I, about the given line, of the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integral.y = √x, y = 0, x = 4, ρ = kx, line: x = 6
Find the Jacobianfor the indicated change of variables. Ifthen the Jacobian of x, y, and z with respect to u, v, and w isx = r cos θ, y = r sin θ, z = z a(x, y, z) d(u, v, w)
Use an iterated integral to find the area of the region bounded by the graphs of the equations.√x + √y = 2, x = 0, y = 0
Use spherical coordinates to find the center of mass of the solid of uniform density.Hemispherical solid of radius r
Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations.z = 9 - x2 - y2 z = 0
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) = k. X y Q: + + a b Z C 1 (a, b, c > 0), x = 0, y = 0, z = 0
Find Ix, Iy, I0, x̅̅ , and y̅̅ for the laminabounded by the graphs of the equations.y = 0, y = b, x = 0, x = a, p = kx
Set up the double integral required to find the moment of inertia I, about the given line, of the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integral.y = √a² - x², y = 0, ρ = ky, line: y = a
Use an iterated integral to find the area of the region bounded by the graphs of the equations.y = x³/², y = 2x
Use spherical coordinates to find the center of mass of the solid of uniform density.Solid lying between two concentric hemispheres of radii r and R, where r < R.
Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations.x² = 9 - y, z² = 9 - y, first octant
Find Ix, Iy, I0, x̅̅ , and y̅̅ for the lamina bounded by the graphs of the equations.y = 4 - x², y = 0, x > 0, ρ = ky
Set up the double integral required to find the moment of inertia I, about the given line, of the lamina bounded by the graphs of the equations. Use a computer algebra system to evaluate the double integral.y = 4 - x², y = 0, ρ = k, line: y = 2
Use a double integral in polar coordinates to find the volume of a sphere of radius a.
Set up the triple integrals for finding the mass and the center of mass of the solid of density ρ bounded by the graphs of the equations.x = 0, x = b, y = 0, y = b, z = 0, z = bρ (x, y, z) = kxy
Use an iterated integral to find the area of the region bounded by the graphs of the equations.2x - 3y = 0, x + y = 5, y = 0
Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations. N || 2 1 + x² + y²¹ z = 0, y = 0, x = 0, y = -0.5x + 1
Use spherical coordinates to find the moment of inertia about the -axis of the solid of uniform density.Solid bounded by the hemisphereand the cone IN VI VI E| + p = cos ,
Use a double integral to find the area of the shaded region. RIN r = 6 cos 0 12345 7 0
Determine the location ofthe 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
Let A be the area of the region in the first quadrant bounded by the line y = 1/2x, the x-axis, and the ellipse 1/9x² + y² = 1. Find the positive number m such that A is equal to the area of the region in the first quadrant bounded by the line y = mx, the y-axis, and the ellipse 1/9x² + y² = 1.
Use an iterated integral to find the area of the region bounded by the graphs of the equations. 2ܐ 6² 1
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 double integral to find the area of the shaded region. [r=2] T 2 1 r = 4 + 3 -0
Find the area of the surface given by z = ƒ(x, y) over the region R.ƒ(x, y) = 25 - x² - y²R = {(x, y): x² + y² ≤ 25}
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) = kx (2,0,3) 73२ X 3 نرا Z 21
Set up the triple integrals for finding the mass and the center of mass of the solid of density ρ bounded by the graphs of the equations.x = 0, x = a, y = 0, y = b, z = 0, z = cρ (x, y, z) = kz
Use spherical coordinates to find the moment of inertia about the -axis of the solid of uniform density.Solid lying between two concentric hemispheres of radii r and R, where r < R

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