Concept explainers
Evaluating a Line Integral In Exercises 23-32, evaluate
along each path. (Hint: If F is conservative, the I
(a)
(b)
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Calculus (MindTap Course List)
- Showing Linear Independence In Exercises 27-30, show that the set of solutions of a second-order linear homogeneous differential equation is linearly independent. {eax,xeax}arrow_forwardEvaluate 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_forwardFind the potential function f for the field F.F = (y - z) i + (x + 2y - z) j - (x + y) k a) f(x, y, z) = x(y + y2) - xz - yz + C b) f(x, y, z) = xy + y2 - x - y + C c) f(x, y, z) = xy + y2 - xz - yz + C d) f(x, y, z) = x + y2 - xz - yz + Carrow_forward
- (a) Evaluate the line integral fc F dr where F(x,y,z) = x i - z j y, k and c is given by r(t) 2t i + 3t j - t2k 1 < t < 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_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
- 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) = x + 3y; C: r(t) = ⟨2 - t, t⟩ , for 0 ≤ t ≤ 2arrow_forwardIn Exercises 1-6, evaluate the integral using the Integration by Parts formula with the given choice of u and d v. j tan- 1 x dx; u = tan- 1 x, d v = dxarrow_forwardLine 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_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_forwardEvaluate the line integral ∫CF→⋅dr→ using the Fundamental Theorem of Line Integrals if F→(x,y)=(4x+4y)i→+(4x+4y)j→and Cis the smooth curve from (−1,1)to (5,6). Enter the exact answer. ∫CF→⋅dr→=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
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning