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Evaluating a Line
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Chapter 15 Solutions
EBK CALCULUS: EARLY TRANSCENDENTAL FUNC
- Line integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The flux line integral of F = ⟨ex - y, ey - x⟩, where C is theboundary of {(x, y): 0 ≤ y ≤ x, 0 ≤ x ≤ 1}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_forwardUse Green's Theorem to evaluate the integral. Assume that the curve C is oriented counterclockwise. 3 In(3 + y) dx - -dy, where C is the triangle with vertices (0,0), (6, 0), and (0, 12) ху 3+y ху dy = 3 In(3 + y) dx - 3+ yarrow_forward
- The figure shows a region R bounded by a piecewise smooth simple closed path C. R (a) Is R simply connected? Explain. (b) Explain why f(x) dx + g(y) dy = 0, where f and g are differentiable functions.arrow_forwardLine integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are oriented counterclockwise.A sketch is helpful. The circulation line integral of F = ⟨x2 + y2, 4x + y3⟩, where Cis the boundary of {(x, y): 0 ≤ y … sin x, 0 ≤ x ≤ π}arrow_forwardApplication of Green's theorem Assume that u and v are continuously differentiable functions. Using Green's theorem, prove that SS'S D Ux Vx |u₁|dA= udv, C Wy Vy where D is some domain enclosed by a simple closed curve C with positive orientation.arrow_forward
- Use Green's theorem to evaluate line integral 2 X √9+ x³ dx + 6xy dy where C is a triangle with vertices (0, 0), (1, 0), and (1, 2) oriented clockwise.arrow_forwardUse Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. $c (5x + sinh y)dy − (3y² + arctan x²) dx, where C is the boundary of the square with vertices (1, 3), (4, 3), (4, 6), and (1, 6). $c (Type an exact answer.) - (3y² + arctan x² (5x + sinh y)dy – nx²) dx dx = (arrow_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ∮C6 y2dx+3 x2dy∮C6 y2dx+3 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.arrow_forward
- ulus 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_forwardSind the absolute maximum and the absolute minimum of f(x,y)=2x– 2xy+y² whose domain is the region defined by 0arrow_forwardUse Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. $c a15x+ 5x+ In 5y)dy - (8y² + arctan x2) dx, where C is the boundary of the square with vertices (0, 5), (2, 5), (2, 7), and (0,7), C $c (5x + In 5y)dy – (8y? + arctanx?) dx = | | C (Type an exact answer.)arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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