Evaluating a Surface
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Chapter 15 Solutions
Calculus (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_forwardulus III |Uni Use Green's Theorem to evaluate the line integral cos (y) dx + x²sin (y) dy along CoS the positively oriented curve C, where C is the rectangle with vertices(0,0), (4, 0), (4, 2) and (0, 2).arrow_forwardUsing Green's Theorem, find the outward flux of F across the closed curve C.F = (-5x + 2y) i + (6x - 9y) j; C is the region bounded above by y = -5x 2 + 250 and below by y=5x2 in the first quadrantarrow_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_forwardApplication of Green's theorem Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that JS D Ur Vy dA= u dv, where D is some domain enclosed by a simple closed curve C with positive orientation.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
- ) 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_forwardClairaut's Theorem Let DC R? be a disk containing the origin and assume that g : D→ R is a function given by g(x, y) = e" (cos y + x sin y). Prove that g(x, y) satisfies the Clairaut Theorem at point (0,0).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
- Clairaut's Theorem Let DCR be a disk containing the origin and assume that q : D → R is a function given by g(x, y) = e" (cos y +x sin y). Prove that g(x, y) satisfies the Clairaut Theorem at point (0, 0).arrow_forwardUse Stokes’ Theorem to evaluate ∫ F*dr where C is oriented counter-clockwise as viewed from above. F(x,y,z) = yi-zj+x2k C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1) Note: The triangle is a portion of the plane x+y+z=1arrow_forwardEvaluate the circulation of G = xyi+zj+7yk around a square of side 9, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Prevs So F.dr-arrow_forward
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