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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

Match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)r(u, v) = ui + ½(u + v)j + vk -2 -2 Z 2 2 2 X y
Find a piecewise smooth parametrization of the path C. (There is more than one correct answer.) 5 4 3 2 (5, 4) 1 2 3 4 5 X
EvaluateS: z = 2/3x³/2, 0 ≤ x ≤ 1, 0 ≤ y ≤ x JsJ G (x - 2y+z) ds.
Find the moments of inertia for a wire of density given by the curve P 1 1 + t
Verify Green’s Theorem by evaluating both integralsfor the given path.C: rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4) y2 dx + x2 dy vP (-) [* = + xP+] we
Show that the value of ∫c F . dr is the same for each parametric representation of C.F(x, y) = yi - xj(a) r₁(θ) = sec θi + tan θj, 0 ≤ θ ≤ π/3(b) r₂(1)=√t + 1i + √tj,0 ≤ t ≤
Verify the Divergence Theorem by evaluatingas a surface integral and as a triple integral. SS ISJ F. NdS
Match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)r(u, v) = ui + 1/4v³j + vk -2 -2 Z 2 2 2 X y
Find the conservative vector field for the potential function by finding its gradient.ƒ(x, y, z) = 2x² + xy + z²
Find the curl of the vector field F.F(x, y, z) = arcsin yi + √1 − x2j + y2k
Verify Stokes's Theorem by evaluating ∫c F . ds = ∫c F . dr as a line integral and as a double integral. F(x, y, z) = (-y + z)i + (x − z)j + (x − y)k S: z = 9x² - y², z ≥ 0
Verify Green’s Theorem by using a computer algebra system to evaluate both integralsfor the given path.C: circle given by x² + y² = 4 J = f (x Јох xey dx + e* dy = С R. ON ам ду dA
Find a piecewise smooth parametrization of the path C. (There is more than one correct answer.) -2-1 2 1 y -2. |x² + y² = 9 1 2 C X
Show that the value of ∫c F . dr is the same for each parametric representation of C.F(x, y) = yi + x²j(a) r1(t) = (2 + t)i + (3 - t)j, 0 ≤ t ≤ 3(b) r₂(w) = (2 + In w)i + (3 - In w)j, 1 ≤
Verify the Divergence Theorem by evaluatingas a surface integral and as a triple integral. SS ISJ F. NdS
Match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)r(u, v) = 2 cos v cos ui + 2 cos v sin uj + 2 sin vk -2 -2 Z 2 2 2 X y
EvaluateS: z = 3 - x - y, first octant SS. xy ds.
Find the conservative vector field for the potential function by finding its gradient.ƒ(x, y, z) = x²eyz
Find ||F|| andsketch several representative vectors in the vector field.F(x, y) = i + j
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x, y) = y i + x² X j
Determine whether the vector field is conservative.F(x, y) = ex(sin yi + cos yj)
Verify Stokes's Theorem by evaluating ∫c F . ds = ∫c F . dr as a line integral and as a double integral. - F(x, y, z) = (-y + z)i + (x − z)j + (x −y)k S: z = √1x² - y² 2
Find a piecewise smooth parametrization of the path C. (There is more than one correct answer.) -2 4 2 -2 + 2 2 C
Verify the Divergence Theorem by evaluatingas a surface integral and as a triple integral. SS ISJ F. NdS
EvaluateS: z = h, 0 ≤ x ≤ 2, 0 ≤ y ≤ √4 - x² SS. xy ds.
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. y F(x, y) = — i — 2/2 ₁ j y x²
Find ||F|| and sketch several representative vectors in the vector field.F(x, y) = yi - 2xj
Match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)r(u, v) = 4 cos ui + 4 sin uj + vk -2 -2 Z 2 2 2 X y
Use a computer algebra system to evaluateS: z = 9 - x², 0 ≤ x ≤ 2, 0 ≤ y ≤ x JSJ xy ds.
Verify Stokes's Theorem by evaluating ∫c F . ds = ∫c F . dr as a line integral and as a double integral. F(x, y, z)= xyzi + yj + zk S: 6x + 6y + z = 12, first octant
Evaluate the line integral along the given path. So C: r(t) = 4ti + 3tj 0 ≤ t ≤ 1 xy ds
Determine whether the vector field is conservative.F(x, y) = 15x²y²i + 10x³yj
Use Green’s Theorem to evaluate the integralfor the given path.C: boundary of the region lying between the graphs of y = x and y = x² - 2x [(y − x) dx + (2x − y) dy -
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Use a line integral to find the area bounded by one arch of the cycloid x(θ) = a(θ - sin θ), y(θ) = a(1 - cos θ), 0 ≤ θ ≤ 2π, as shown in the figure. 2a y Σπα
Find ||F|| and sketch several representative vectors in the vector field.F(x, y, z) = Зуј
Determine whether the vector field is conservative. F(x, y) = 1/2 (yi y² (yi + xj)
Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. r(u, v) = ui + vj + -k 2
Evaluate the line integral along the given path. 3(x - y) ds Jc C: r(t) = ti + (2 - t)j 0 ≤t≤2
Verify Stokes's Theorem by evaluating ∫c F . ds = ∫c F . dr as a line integral and as a double integral. F(x, y, z) = z²i + x²j + y²k S: z = y², 0≤x≤ a, 0≤ y ≤ a
Use a computer algebra system to evaluateS: z = 1/2xy, 0 ≤ x ≤ 4, 0 ≤ y ≤ 4 JSJ xy ds.
Use Green’s Theorem to evaluate the integralfor the given path.C: x = 2 cos θ, y = sin θ [(y − x) dx + (2x − y) dy -
Use a line integral to find the area bounded by the two loops of the eight curve 1 x(t) = si sin 2t, y(t) = sin t, 0 ≤ t ≤ 2π
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.F(x, y) = (xy² - x²)i + (x²y + y²) j
Find ||F|| and sketch several representative vectors in the vector field.F(x, y) = yi + xj
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Determine whether the vector field is conservative.F(x, y, z) = y ln zi - xln zj + xy/z k
Use Green’s Theorem to calculate the work done by the force F on a particle that is moving counterclockwise around the closed path C.F(x, y) = (ex - 3y)i + (ey + 6x)jC: r 2 cos θ
Find the flux of F through S,where N is the upward unit normal vector to S. JsJ F. NdS
Find the total mass of the wire with density ρ.r(t) = t²i + 2tj, ρ(x, y) = 3/4y, 0 ≤ t ≤ 1
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. x F(x, y, z) = arctani + In√x² + y²j + k y C: triangle with vertices (0, 0, 0), (1,
Use a computer algebra system to evaluate S.S (x ². s. (x² - 2xy) ds.
Evaluate the line integral along the given path. 2xyz, ds C: r(t) = 12ti + 5tj + 84tk 0 ≤t≤ 1
The force field F(x, y) = (3x²y²)i + (2x³y)j is shown in the figure below. Three particles move from the point (1, 1) to the point (2, 4) along different paths. Explain why the work done is the
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Use Green’s Theorem to evaluate the integralfor the given path.C: boundary of the region lying inside the semicircle y = √25 - x² and outside the semicircle y = √9 - x² [(y − x) dx + (2x
Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. r(u, v) = 3 cos v cos ui + 3 cos v sin uj + 5
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = z²i + 2xj + y²k S: z = 1x² - y², z ≥ 0
Use a computer algebra system to graph several representative vectors in the vector field. F(x, y) = (2xyi + y²j)
Determine whether the vector field is conservative.F(x, y, z) = sin yzi + xz cos yzj + xy sin yz k
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y) = 2xyi + x²j(a) r1(t) = ti + t²j,0 ≤ t ≤ 1(b) r₂(t) = ti + t³j, 0 ≤ t
(a) Find a parametrization of the path C(b) Evaluatealong C.C: line segment from (0, 0) to (1, 1) JC (x² + y²) ds
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = 2u cos vi + 2u sin vj + u¹k 0 ≤u≤ 1, 0 ≤ y ≤ 2π
Use Green’s Theorem to evaluate the line integral. 1² 2xy dx + (x + y) dy C: boundary of the region lying between the graphs of y = 0 and y = 1 - x²
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.F(x, y, z) = (4xy + z²)i + (2x² + 6yz)j + 2xzk
Let S be a smooth oriented surface with normal vector N, bounded by a smooth simple closed curve C. Let v be a constant vector, and prove that ff (2v. S. (2v N) ds = = (v x r). dr. с
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Find the mass of the surface lamina of density ρ.S: 2x + 3y + 6z = 12, first octant, ρ(x, y, z) = x² + y²
Determine whether the vector field is conservative. If it is, find a potential function for the vector field. F(x, y, z) = yzi - xzjxyk y²z²
Use a computer algebra system to graph several representative vectors in the vector field. F(x, y) = (2y x, 2y + x) -
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 4xzi + yj + 4xyk S: z = 9x² - y², z 20
(a) Find a parametrization of the path C(b) Evaluatealong C.C: line segment from (0, 0) to (2, 4) JC (x² + y²) ds
Use Green’s Theorem to evaluate the line integral. Ly² dx y² dx + xy dy C: boundary of the region lying between the graphs of y = 0, y = √√√x, and x = 9
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = 2 cos v cos ui + 4 cos v sin uj + sin vk 0 ≤ u≤ 2π, 0≤ v ≤ 2π
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y) = yexyi + xexyj(a) r₁(t) = ti - (t − 3)j, 0 ≤ t ≤ 3(b) The closed path
How does the area of the ellipse compare with the magnitude of the work done by the force fieldon a particle that moves once around the ellipse (see figure)? บร a + 2 1
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Use a computer algebra system to graph several representative vectors in the vector field. F(x, y, z)= = xi + yj + zk √x² + y² + z² 2
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = z²i + yj + zk S: z = √√√4x² - y²
Find the mass of the surface lamina of density ρ.S: z = √a² - x² - y², ρ(x, y, z) = kz
(a) Find a parametrization of the path C(b) Evaluatealong C.C: counterclockwise around the circle x² + y² = 1 from (1, 0) to (0, 1) JC (x² + y²) ds
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Evaluateƒ(x, y) = y + 5S: r(u, v) = ui + vj + 2vk, 0 ≤ u ≤ 1, 0 ≤ v ≤ 2 SS f (x, y) ds.
Determine whether the vector field is conservative. If it is, find a potential function for the vector field.F(x, y, z) = sin z(yi + xj + k)
Use Green’s Theorem to evaluate the line integral. Locx (x² - y²) dx + 2xy dy Jc C: x² + y² = 16
Find (a) Thedivergence of the vector field F (b) The curl of the vectorfield FF(x, y, z) = x²i + xy²j + x²zk
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y) = yi - xj(a) r1(t) = ti + tj, 0 ≤ t ≤ 1 (b) r₂(t) = ti + t²j, 0 ≤
Use a computer algebra system to graph several representative vectors in the vector field. F(x, y, z)=(x, y, z)
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = x²i + z²j - xyzk S: z = √√√4x² - y² -
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v): = 2 sinh u cos vi + sinh u sin vj + cosh uk 0 ≤u≤ 2, 0 ≤ y ≤ 2π
(a) Find a parametrization of the path C(b) Evaluatealong C.C: counterclockwise around the circle x² + y² = 4 from (2, 0) to (0, 2) JC (x² + y²) ds
Use Green’s Theorem to evaluate the line integral. (x² - y²) dx + 2xy dy Jc C: r = 1 + cos 0
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)F(x, y) = xy²i + 2x²yj(a) r1(t) = ti + 1/tj, 1 ≤ t ≤ 3(b) r₂(t) = (t + 1)i -
Evaluate S: r(u, v) = ui + vj + 2vk, 0 ≤ u ≤ 1, 0 ≤ v ≤ SS f (x, y) ds.
Find (a) The divergence of the vector field F (b) The curl of the vector field FF(x, y, z) = y2j - z²k
Use the Divergence Theorem to evaluateand find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results.
Use a computer algebra system to graph the surface represented by the vector-valued function. r(u, v) = 2u cos vi + 2u sin vj + vk 0 ≤u≤ 1, 0 ≤ v ≤ 3π
Use Stokes's Theorem ∫c F . dr In each case, C is oriented counterclockwise as viewed from above. F(x, y, z) = −ln\ x + y2i+ arctan-j+k S: z = 9 2x - 3y over r = 2 sin 20 in the first
Find the value of the line integralIf F is conservative, the integration may be easier on an alternative path.)(a)(b)(c)(d) C F. dr.
(a) Find a parametrization of the path C(b) Evaluatealong C.C: x-axis from x = 0 to x = 1 √√(x + 4 + 4√y) ds C

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