Evaluating a Line
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Evaluating a Line Integral for Different Parametrizations In Exercises 1–4, show that the value of
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(b).
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Chapter 15 Solutions
Calculus: Early Transcendental Functions (MindTap Course List)
- Using 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_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_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
- Evaluate 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_forwardStokes’ 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.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 line segment y = 10 - 2x from (1, 8) to (3, 4)arrow_forward
- Work in a hyperbolic field Consider the hyperbolic force field F = ⟨y, x⟩ (the streamlines are hyperbolas) and the three paths shown in the figure. Compute the work done in the presence of F on each of the three paths. Does it appear that the line integral ∫C F ⋅ T ds is independent of the path, where C is any path from (1, 0) to (0, 1)?arrow_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_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 = ⟨y, x⟩; C is the circle of radius 8 centered at the origin orientedcounterclockwise.arrow_forward
- Using green's theorem, evaluate the line integral xy^2dx + (1-xy^3)dyarrow_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⟩; C is the circle of radius 4 centered at the origin orientedcounterclockwise.arrow_forwardUse 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).arrow_forward
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