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Verifying Green’s TheoremIn Exercises 5–8, verify Green’s Theorem by evaluating both
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Chapter 15 Solutions
Calculus (MindTap Course List)
- Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨-y, x⟩ on the parabola y = x2 from (0, 0) to (1, 1)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_forwardUsing Cauchy's Theorem calculate the following integral and the singular points of the function, where C: z(t) = 3*cost(t) + i*(3+ 3*sin(t)) 0 < t < 2πarrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C4 y2dx+6 x2dy∮C4 y2dx+6 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. ∮C4 y2dx+6 x2dy=arrow_forwardA. State Green’s Theorem for a vector field F = Pi + Qj defined on a plane region R bounded by a closed curve C. B. Verify Green’s Theorem by evaluating both integrals when F = yi − xj and R is the plane region bounded by a circle C is of radius a.arrow_forwardUsing green's theorem, evaluate the line integral xy^2dx + (1-xy^3)dyarrow_forward
- Line integrals of vector fields in the plane Given the followingvector fields and oriented curves C, evaluate ∫C F ⋅ T ds. F = ⟨x, y⟩ on the parabola r(t) = ⟨4t, t2⟩ , for 0 ≤ t ≤ 1arrow_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_forwardLine integrals of vector fields on closed curves Evaluate ∮C F ⋅ dr for the following vector fields and closed oriented curves C by parameterizing C. If the integral is not zero, give an explanation. F = ⟨x, y, z⟩; C: r(t) = ⟨cos t, sin t, 2⟩ , for 0 ≤ t ≤ 2πarrow_forward
- Evaluating 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_forwardEvaluating an iterated integral Evaluate V = ∫10 A(x) dx, whereA(x) = ∫20 (6 - 2x - y) dy.arrow_forwardWork integrals Given the force field F, find the work required to move an object on the given oriented curve. F = ⟨y, x⟩ on the parabola y = 2x2 from (0, 0) to (2, 8)arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning