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Using the Fundamental Theorem of Line
F (x, y) =
C: line segment from
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Multivariable Calculus
- Using Stokes' theorem, solve the line integral of G(x, y, z) - (1, x + yz, xy-√z) around the boundary of surface S, which is given by the piece of the plane 3x + 2y + z = 1 where x, y, and z all ≥ 0.arrow_forwardUse Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwise indicated. ?C(lnx+y)dx−x2dy where C is the rectangle with vertices (1, 1), (3, 1), (1, 4), and (3, 4)arrow_forwarduse Green 's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwise indicated. ∮C y2 dx + x2 dy, where C is the boundary of the square that is given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1arrow_forward
- A. State the F undamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = xy + 2yz + 3zx and F = grad f. Find the line integral of F along the line C with parametric equations x = t, y = t, z = 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardA. State the Fundamental Theorem of Calculus for Line Integrals. B. Let f(x, y, z) = x^2 + 2y^2 + 3z^2 and F = grad f. Find the line integral of F along the line C with parametric equations x = 1 + t, y = 1 + 2t, z = 1 + 3t, 0 ≤ t ≤ 1. You must compute the line integral directly by using the given parametrization. C. Check your answer in Part B by using the Fundamental Theorem of Calculus for Line Integrals.arrow_forwardEvaluate the line integrals using the Fundamental Theorem of Line Integrals: ∫c (yi+xj)*dr Where C is any path from (0,0) to (2,4).arrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C 3 y2 dx+3 x2 dy, where CC is the square with vertices (0,0)(0,0), (2,0)(2,0), (2,2)(2,2), and (0,2)(0,2) oriented counterclockwise.arrow_forwardUse Stokes' Theorem to calculate the line integral of the vector function F = (y, z, x) along the curve C, which is obtained by the intersection of the surfaces x + y = 2 and x² + y² + z² = 2 ( x + y).arrow_forwardUse Stokes's Theorem to evaluate integral C of F · dr. In this case, C is oriented counterclockwise as viewed from above. F(x, y, z) = z2i + 2xj + y2k S: z = 1 − x2 − y2, z ≥ 0arrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C6 y2dx+3 x2dy∮C6 y2dx+3 x2dy, where CC is the square with vertices (0,0)(0,0), (3,0)(3,0), (3,3)(3,3), and (0,3)(0,3) oriented counterclockwise.arrow_forwardLet the curve C be the line segment from (0, 0) to (3, 1). Let F = ⟨2x-y, 4y-x⟩ Calculate the integral ∫c F· dr = ∫c (2x-y)dx + (4y-x)dy in two different ways:(a) Parameterize the curve C and compute the integral directly. (b) Use the Fundamental Theorem of Line Integrals.arrow_forwardEvaluate the line integral ∮C F ⋅ dr using Stokes’ Theorem. Assume C has counterclockwise orientation. F = ⟨xz, yz, xy⟩; C is the circle x2 + y2 = 4 in the xy-plane.arrow_forward
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