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mathematics
calculus 10th edition

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

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cylinder r(u, v) = a cos ui + a sin uj + vk, where 0 ≤ u ≤ 27 and 0 ≤ y ≤ b
Describe an orientable surface.
Evaluate the integral using the Fundamental Theorem of Line Integrals. So C: smooth curve from (0, 0, 0) to (1, 3, 2) 2xyz dx + x²z dy + x²y dz
Determine whether the work done along the path C is positive, negative, or zero. Explain. 1 C X
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.F(x, y) = ex(cos yi - sin yj)
Define a flux integral and explain how it is evaluated.
Find the area of the surface over the given region. Use a computer algebra system to verify your results. The sphere r(u, v) = a sin u cos vi + a sin u sin vj + a cos uk, where 0 ≤ u ≤ 7 and 0 ≤ y ≤ 2π
State the Fundamental Theorem of Line Integrals.
Use Green’s Theorem to explain whywhere ƒ and g are differentiable functions and C is a piecewise smooth simple closed path (see figure). fra C f(x) dx + g(y) dy = 0
Is the surface shown in the figure orientable? Explain why or why not. Double twist
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x, y) = 2xi + 2yj (x² + y²)²
Evaluate the integral using the Fundamental Theorem of Line Integrals. Lydx 1 = dz Z C: smooth curve from (0, 0, 1) to (4, 4, 4) y dx + x dy +
LetFind the value of the line integral ∫c F . dr.(a)(b)(c)(d) F(x, y) y x² + y² i X x² +
Determine whether the work done along the path C is positive, negative, or zero. Explain. X
Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cone r(u, v): where 0 ≤ u≤ b and 0 ≤ y ≤ 2π = au cos vi + au sin vj + uk,
(a) Use a computer algebra system to graph the vector-valued functionThis surface is called a Möbius strip.(b) Is the surface orientable? Explain why or why not.(c) Use a computer algebra system to graph the space curve represented by r(u, 0). Identify the curve.(d) Construct a Möbius strip by
Let R be the region inside the circle x = 5 cos θ, y = 5 sin θ and outsidethe ellipse x = 2 cos θ, y = sin θ. Evaluate the line integralwhere C = C₁ + C₂ is the boundary of R, as shown in the figure. √ (e-x²/2y) dx + (e-²/2 + x) dy -
Consider the force field shown in the figure.(a) Give a verbal argument that the force field is not conservative because you can identify two paths that require different amounts of work to move an object from (-4, 0) to (3, 4). Of the two paths, which requires the greater amount of work?(b) Give a
What does it mean that a line integral is independent of path? State the method for determining whether a line integral is independent of path.
Determine whether the work done along the path C is positive, negative, or zero. Explain. - X
Let R be the region inside the ellipse x = 4 cos θ, y = 3 sin θ and outsidethe circle x = 2 cos θ, y = 2 sin θ. Evaluate the line integralwhere C = C₁ + C₂ is the boundary of R, as shown in the figure. [ (3x²y + 1) dx + (x³ + 4x) dy C
Find the area of the surface over the given region. Use a computer algebra system to verify your results. The torus r(u, v) = (a + b cos v) cos ui + (a + bcos v) sin uj + b sin vk, where a > b, 0 ≤ u ≤ 27, and 0 ≤ y ≤ 2π
Find curl F for the vector field at the given point.F(x, y, z) = xyzi + xyzj + xyzk; (2, 1, 3)
Find the area of the surface over the given region. Use a computer algebra system to verify your results. The surface of revolution r(u, v) = √√u cos vi+ √u sin vj + uk, where 0 ≤ u ≤ 4 and 0 ≤ y ≤ 2π
Determine whether the work done along the path C is positive, negative, or zero. Explain. 1c/ X
Use Green’s Theorem to evaluate the line integral. Lydx y dx + 2x dy C: boundary of the square with vertices (0, 0), (0, 1), (1, 0), and (1, 1)
Find the area of the surface over the given region. Use a computer algebra system to verify your results. The surface of revolution sin u sin vk, where 0 ≤ u≤7 and 0 ≤ y ≤ 2π r(u, v) = sin u cos vi + uj +
Find curl F for the vector field at the given point.F(x, y, z) = x²zi − 2xzj + yzk; (2, — 1, 3)
Consider the region bounded by the x-axis and one arch of the cycloid with parametric equations x = a(θ - sin θ) and y = a(1- cos θ). Use line integrals to find (a) The area of the region(b) The centroid of the region
Consider the force field shown in the figure. Is the force field conservative? Explain why or why not. + → X
For each given path, verify Green’s Theorem by showing thatFor each path, which integral is easier to evaluate? Explain.(a) C: triangle with vertices (0, 0), (4, 0), and (4,4) (b) C: circle given by x² + y² = 1 ƏN [xax+8& = [x-Ma y2 dx + x2 dy ду C R. dA.
(a) Let C be the line segment joining (x₁, y₁) and (x₂, y₂). Show that ∫c -y dx + x dy = X₁Y2 - X₂y₁.(b) Let (x₁, y₁), (x₂, y₂),..., (xn, yn) be the vertices of a polygon. Prove that the area enclosed is 1/[(x₁₂x₂y₁) + (x₂ V3 — X3Y₂) +・・・
Find curl F for the vector field at the given point.F(x, y, z) = ex sin yi - ex cos yj; (0, 0, 1)
Use Green’s Theorem to evaluate the line integral. Se xy dx + (x² + y²) dy C: boundary of the square with vertices (0, 0), (0, 2), (2, 0), and (2, 2)
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If C₁, C₂, and C3 have the same initial and terminal points and . Sc, F. dr₁ = Sc₂ F dr₂, then Sc, Fdr₁ = Sc, F. dr3.
Use a computer algebra system to find the curl F for the vector field. F(x, y, z) = arctan X i + In√x² + y² j + k
Evaluate ∫c F . dr each curve. Discussthe orientation of the curve and its effect on the value of theintegral.F(x, y) = x²i + xyj(a) r1(t) = 2ti + (t− 1)j, 1 ≤ t ≤ 3(b) r₂(t) = 2(3 − t)i + (2 − t)j, 0 ≤ t ≤ 2
Demonstrate the property thatregardless of the initial and terminal points of C, where the tangent vector r'(t) is orthogonal to the force field F. So C F. dr = 0
Use the result of Exercise 47(b) to find the area enclosed by the polygon with the given vertices.(a) Pentagon: (0, 0), (2, 0), (3, 2), (1,4), and (-1, 1)(b) Hexagon: (0, 0), (2, 0), (3, 2), (2, 4), (0, 3), and (-1, 1)Data from in Exercise 47(b)(b) Let (x₁, y₁), (x₂, y₂),..., (xn, yn) be
Find curl F for the vector field at the given point.F(x, y, z) = e-xyz (i + j + k); (3, 2, 0)
Use Green’s Theorem to evaluate the line integral. So xy² dx + x²y dy C: x = 4 cos t, y = 4 sin t
Evaluate ∫c F . dr each curve. Discuss the orientation of the curve and its effect on the value of the integral.F(x, y) = x²yi + xy³/²j(a) r1(t) = (t + 1)i + t²j, 0 ≤ t ≤ 2(b) r₂(t) = (1 + 2 cos t)i + (4 cos² t)j, 0 ≤ t ≤ π/2
Use Green’s Theorem to evaluate the line integral. [ (x² - y²) dx + 2xy dy C: x² + y² = a² 2
Demonstrate the property thatregardless of the initial and terminal points of C, where the tangent vector r'(t) is orthogonal to the force field F. So C F. dr = 0
Define a parametric surface.
Use a computer algebra system to find the curl F for the vector field. F(x, y, z) = yz y-z i + XZ, X-Z j+ ху x - y -k
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If F = yi + xj and C is given by r(t) = (4 sin t)i + (3 cos t)j, for 0 ≤ t ≤ π, then ∫c F . dr = 0.
Give the double integral that yields the surface area of a parametric surface over an open region D.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If F is conservative in a region R bounded by a simple closed path and C lies within R, then ∫c F. dr is independent of path.
Prove the identity, where R is a simply connected region with boundary C. Assume that the required partial derivatives of the scalar functions ƒ and g are continuous. The expressions DNƒ and DNg are the derivatives in the direction of the outward normal vector N of C, and are defined by DNƒ =
Demonstrate the property thatregardless of the initial and terminal points of C, where the tangent vector r'(t) is orthogonal to the force field F. So C F. dr = 0
Use Green’s Theorem to evaluate the line integral. la xy dx + x² dy C: boundary of the region between the graphs of y = x² and y = 1
Use a computer algebra system to find the curl F for the vector field. F(x, y, z) = √√x² + y² + z² (i + j + k)
The figures below are graphs of r(u, v) = ui+ sin u cos vj + sin u sin vk, where 0 ≤ u ≤ π/2 and 0 ≤ v ≤ 2π. Match eachof the graphs with the point in space from whichthe surface is viewed. The points are (10, 0, 0),(-10, 10, 0), (0, 10, 0), and (10, 10, 10).(a)(b)(c)(d) N
Demonstrate the property thatregardless of the initial and terminal points of C, where the tangent vector r'(t) is orthogonal to the force field F. So C F. dr = 0
Use a computer algebra system to find the curl F for the vector field.F(x, y, z) = sin(x - y)i + sin(y - z)j + sin(z − x)k
A function ƒ is called harmonic whenProve that if ƒ is harmonic, thenwhere C is a smooth closed curve in the plane. 时时 + ox² dy = 0.
Prove the identity, where R is a simply connected region with boundary C. Assume that the required partial derivatives of the scalar functions ƒ and g are continuous. The expressions DNƒ and DNg are the derivatives in the direction of the outward normal vector N of C, and are defined by DNƒ =
Use Green’s Theorem to evaluate the line integral. Sex C: x2/3 + y2/3 = 1 y² dx + x4/3 dy
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If F = Mi + Nj and ∂m/∂x =∂N/∂y, then F is conservative.
Astroidal Sphere An equation of an astroidal sphere in x, y, and z isA graph of an astroidal sphere is shown below. Show that this surface can be represented parametrically by x2/3 + 2/3 + 2/3 = a²/³.
Evaluate the line integral along the path Cgiven by x = 2t, y = 10t, where 0 ≤ t ≤ 1. Jc (x + 3y²) dy
Use a computer algebra system to graph three views of the graph of the vector-valued functionfrom the points (10, 0, 0), (0, 0, 10), and (10, 10, 10). r(u, v) = u cos vi + u sin vj + vk, 0 ≤ u ≤ π, 0≤ V≤ T
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = sec u cos vi + (1 + 2 tan u) sin vj + 2uk 0 ≤u≤7, 3 0≤ y ≤ 2π
Determine whether the vector field F is conservative. If it is, find a potential function for the vector field.F(x, y, z) = xy²z²i + x²yz²j + x²y²zk
Let F = Mi + Nj, where M and N have continuous first partial derivatives in a simply connected region R. Prove that if C is simple, smooth, and closed, and Nx = My, then ∫c F . dr = 0.
The kinetic energy of an object moving through a conservative force field is decreasing at a rate of 15 units per minute. At what rate is the potential energy changing?
Determine whether the vector field F is conservative. If it is, find a potential function for the vector field.F(x, y, z) = y²z³i + 2xyz³j + 3xy²z²k
Evaluate the line integral along the path C given by x = 2t, y = 10t, where 0 ≤ t ≤ 1. facx Jc (x + 3y²) dx
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = e¯“/4 cos vi + e−"/4 sin vj + k 0 ≤ u≤ 4, 0 ≤ y ≤ 2π
Evaluate the line integral along the path C given by x = 2t, y = 10t, where 0 ≤ t ≤ 1. So с xy dx + y dy
Let(a) Show that(b) Let r(t) = cos ti + sin tj for 0 ≤ t ≤ π. Find ∫c F . dr.(c) Let r(t) = cos ti - sin tj for 0 ≤ t ≤ π. Find ∫c F . dr.(d) Let r(t) = cos ti + sin tj for 0 ≤ t ≤ 2π. Find ∫c F . dr.(e) Show that y F(x, y) = x² + y² X 2 x² + y2j.
Evaluate the surface integral ∫s∫ z dS over the surface S: - r(u, v) = (u + v)i + (u − v)j + sin vk where 0 ≤ u ≤ 2 and 0 ≤ y ≤ TT.
Evaluate the line integral along the path C given by x = 2t, y = 10t, where 0 ≤ t ≤ 1. I с (3y - x)dx + y² dy Зу
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola xy = 1 and both branches of the hyperbola xy = -1. (A set S in the plane is called convex if for any two points in S the line segment connecting them is contained in S.)
Determine whether the vector field F is conservative. If it is, find a potential function for the vector field. XZ F(x, y, z) = 1-J+k y y y
Find a vector-valued function for the hyperboloidand determine the tangent plane at (1, 0, 0). [ = 2 - x² + y²
Evaluate the integralalong the path C.C: y-axis from y = 0 to y = 2 (((2x - y) (2x - y) dx + (x + 3y) dy C
Determine whether the vector field F is conservative. If it is, find a potential function for the vector field.F(x, y, z) = sin zi + sin xj + sin yk
Use a computer algebra system to graph the surface S and approximate the surface integralwhere S is the surface SSG (x + y) ds
Determine whether the vector field F is conservative. If it is, find a potential function for the vector field.F(x, y, z) = yezi + zexj + xeyk
Evaluate the integralalong the path C.C: x-axis from x = 0 to x = 5 (((2x - y) (2x - y) dx + (x + 3y) dy C
Graph and find the area of one turn of the spiral ramp r(u, v) = u cos vi + u sin vj + 2vk where 0 ≤ u ≤ 3 and 0 ≤ y ≤ 2TT.
Determine whether the vector field F is conservative. If it is, find a potential function for the vector field. F(x, y, z) = X y i + x² + y² x² + y² j + k
A cone-shaped surface lamina S is given byAt each point on S, the density is proportional to the distance between the point and the z-axis.(a) Sketch the cone-shaped surface.(b) Find the mass m of the lamina. z = a(a - √√x² + y²), 0≤z ≤a². 2
The surface of the dome on a new museum is given byr(u, v) = 20 sin u cos vi + 20 sin u sin vj + 20 cos ukwhere 0 ≤ u ≤ π/3,0 ≤ v ≤ 2π, and r is in meters. Find the surface area of the dome.
Verify the Divergence Theorem by evaluatingas a surface integral and as a triple integral. SS s. F. NdS
Verify the Divergence Theorem by evaluatingas a surface integral and as a triple integral. SS s. F. NdS
Evaluate the integralalong the path C.C: line segments from (0, 0) to (3, 0) and (3, 0) to (3, 3) (((2x - y) (2x - y) dx + (x + 3y) dy C
Evaluate the integralalong the path C.C: line segments from (0, 0) to (0, -3) and (0, -3) to (2, -3) (((2x - y) (2x - y) dx + (x + 3y) dy C
Let f be a nonnegative function such that ƒ'is continuous over the interval [a, b]. Let S be the surface ofrevolution formed by revolving the graph of ƒ, wherea ≤ x ≤ b, about the x-axis. Let x = u, y = ƒ(u) cos v, andz = ƒ(u) sin v, where a ≤ u ≤ b and 0 ≤ v ≤ 2π. Then,S is
The parametric equationswhere - π ≤ u ≤ π and -π ≤ v ≤ π represent the surface shown below. Try to create your own parametric surface using a computer algebra system. x = 3 + sin u[7 - cos(3u2v) - 2 cos(3u + v)] y = 3 + cos u[7 - cos(3u2v) - 2 cos(3u + v)] z = sin(3u2v) + 2 sin(3u + v)
Find the divergence of the vector field F.F(x, y) = x²i + 2y²j
Verify Stokes’s Theorem by evaluatingas a line integral and as a double integral.F(x, y, z) = (cos y + y cos x)i + (sin x - x sin y)j + xyzkS: portion of z = y2 over the square in the xy-plane with vertices (0, 0), (a, 0), (a, a), and (0, a)N is the upward unit normal vector to the surface. JC F.
Find the divergence of the vector field F.F(x, y) = xexi + yeyj
Evaluate the integralalong the path C.C: arc on y = 1 - x² from (0, 1) to (1, 0) (((2x - y) (2x - y) dx + (x + 3y) dy C
The surface shown in the figure is called a Möbius strip and can be represented by the parametric equationswhere -1 ≤ u ≤ 1, 0 ≤ v ≤ 2π, and a = 3. Try to graph other Möbius strips for different values of a using a computer algebra system. = (a + X = a + u cos u cos = (a + ₁ Cos V, y
Find the divergence of the vector field F.F(x, y, z) = sin xi + cos yj + z²k
Evaluate the integralalong the path C.C: arc on y = x³/2 from (0, 0) to (4, 8) (((2x - y) (2x - y) dx + (x + 3y) dy C
Verify Stokes’s Theorem by evaluatingas a line integral and as a double integral. JC F. dr

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