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Using the Fundamental Theorem of Line
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Calculus (MindTap Course List)
- A. 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_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forwardEvaluating line integrals Use the given potential function φ of the gradient field F and the curve C to evaluate the line integral ∫C F ⋅ dr in two ways.a. Use a parametric description of C and evaluate the integral directly.b. Use the Fundamental Theorem for line integrals. φ(x, y, z) = xy + xz + yz; C: r(t) = ⟨t, 2t, 3t⟩ , for 0 ≤ t ≤ 4arrow_forward
- Use 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 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_forwardUsing Stokes’ Theorem to evaluate a line integral Evaluate the lineintegral ∮C F ⋅ dr, where F = z i - z j + (x2 - y2) k and C consists of the three line segments that bound the plane z = 8 - 4x - 2y in the first octant, oriented as shown.arrow_forward
- Using the surface integral in stokes theorem find the circulation of the field F around the curve Carrow_forwardCheck Stokes' Theorem, evaluating the two integrals of the statement, to F(x, y, z) = (y, −x, 0), the paraboloid S : z = x2 + y2, with 0 ≤ z ≤ 1, and n pointing out of S. Answer is 1/2arrow_forwardEvaluate 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_forward
- Let 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_forwardA) Evaluate the given line integral directly. B) Evaluate the given line integral by using Green's theorem.arrow_forwardEvaluate C F · dr using the Fundamental Theorem of Line Integrals. F(x, y, z) = 2xyzi + x2zj + x2yk C: smooth curve from (0, 0, 0) to (1, 3, 2)arrow_forward
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