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Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 12T

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Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. If needed, you can use GeoGebra to graph the region. Also if the evaluation
of the double integral is very complicated, you can use the help of your computer - for example you can use wolfram alpha or symbolab.
F = (x - y) i + (x + y) j; C is the triangle with vertices at (0, 0), (8, 0), and (0, 7)
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Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. If needed, you can use GeoGebra to graph the region. Also if the evaluation
of the double integral is very complicated, you can use the help of your computer - for example you can use wolfram alpha or symbolab.
F = xy i + xj; C is the triangle with vertices at (0, 0), (8, 0), and (0, 4)
O 112
80
3

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Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. If needed, you can use GeoGebra to graph the region. Also if the evaluation of the double integral is very complicated, you can use the help of your computer - for example you can use wolfram alpha or symbolab. F = (x 2 + y 2) i + (x - y) j; C is the rectangle with vertices at (0, 0), (6, 0), (6, 9), and (0, 9)
Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. If needed, you can use GeoGebra to graph the region. Also if the evaluation of the double integral is very complicated, you can use the help of your computer - for example you can use wolfram alpha or symbolab. F = (x 2 + y 2) i + (x - y) j; C is the rectangle with vertices at (0, 0), (6, 0), (6, 9), and (0, 9) a) 540 b) 432 c) -432 d) 0
StokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).
Use Green's Theorem to evaluate the line integral. Assume curve is oriented counterclockwise. C is the boundary of the square with vertices (3,1), (6,1), (6,4), and (3,4) Please refer to attached image.
The area pab of the ellipse x2/a2 + y2/b2 = 1 can be found by integrating the function ƒ(x, y) = 1 over the region bounded by the ellipse in the xy-plane. Evaluating the integral directly requires a trigonometric substitu-tion. An easier way to evaluate the integral is to use the transforma-tion x = au, y = by and evaluate the transformed integral over the disk G: u2 + y2<= 1 in the uy-plane. Find the area this way.
Using green's theorem, evaluate the line integral xy^2dx + (1-xy^3)dy
Stokes’ Theorem for evaluating line integrals Evaluate theline integral ∮C F ⋅ dr by evaluating the surface integral in Stokes’Theorem with an appropriate choice of S. Assume C has a counterclockwiseorientation. F = ⟨y, xz, -y⟩; C is the ellipse x2 + y2/4 = 1 in the plane z = 1.
Use Stokes' Theorem to evaluate (integral) F (dot) dr. C is oriented counterclockwise as viewd from above. F (x,y,z) = < 2z + x , y - z , x + y > C is the triangle with vertices (1,0,0), (0,1,0), (0,0,1).
(a) Sketch the region of integration R in the xy - plane and sketch the region G in the uv - plane using the coordinate transformation x = 2u and y = 2u + 4v.
Let the solid S be bounded by an elliptic paraboloid z = 9-x2-y2 and planes x = 0, y = x & z = 0. Set up a double integral in rectangular coordinates that will give the value of the volume of solid S. And, this time, using polar coordinates, use double integrals to evaluate the volume of solid S.
se the Divergence Theorem to evaluate S F · N dS and 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. F(x, y, z) = x3i + x2yj + x2eyk S: z = 4 − y, z = 0, x = 0, x = 3, y = 0
Suppose C is the boundary of region R = {(x, y): x2 ≤ y ≤ x ≤ 1}, oriented counterclockwise (see figure); let F = ⟨1, x⟩. a. Compute the two-dimensional curl of F and determine whetherF is irrotational.b. Find parameterizations r1(t) and r2(t) for C1 and C2, respectively.c. Evaluate both the line integral and the double integral in the circulationform of Green’s Theorem and check for consistency.d. Compute the two-dimensional divergence of F and use the fluxform of Green’s Theorem to explain why the outward flux is 0.

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