Concept explainers
Using Green's Theorem to Verify a Formula In Exercises 33 and 34, use Green’s Theorem to verify the line
The centroid of the region having area A bounded by the simple closed path C has coordinates
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Calculus (MindTap Course List)
- Find the area between the curves in Exercises 1-28. x=2, x=1, y=2x2+5, y=0arrow_forward) Using Green's theorem, convert the line integral f.(6y² dx + 2xdy) to a double integral, where C is the boundary of the square with vertices ±(2, 2) and ±(2,-2). ( do not evaluate the integral)arrow_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 5y'dx + 4x°dy, where Cis the square with vertices (0, 0), (1, 0). (1, 1), and (0, 1) oriented counterclockwise. f 5y'dx + 4x°dy iarrow_forward
- Use Green's Theorem to evaluate the following integral Let² dx + (5x + 9) dy Where C is the triangle with vertices (0,0), (11,0), and (10, 9) (in the positive direction).arrow_forwardUse Green's theorem to calculate the integral - dx + dy, where C is a rectangle with vertices (–1,1), (2, 1), (2, 5), and (–1,5), oriented counterclockwise.arrow_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. P y*dx + 2x²dy, where C is the square with vertices (0, 0), (3, 0), (3, 3), and (0, 3) oriented counterclockwise. P y'dx + 2x*dy :arrow_forward
- 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_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_forwardUse Green's Theorem to evaluate the line integral Lu+o") dx + (3x+cosy) dy where C is the triangle with vertices (0, 0), (0, 2) and (2, 2) oriented counterclockwise. un O 6 O 8 O 14 O 4 O 10 O 12arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,