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Evaluating a Line
C: boundary of the region lying inside the rectangle with vertices
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CALC.,EARLY TRANSCEND..(LL)-W/WEBASSIGN
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- 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_forwardUse Green's Theorem to evaluate the line integral. Assume the curve is oriented counterclockwise. (9x + In 9y)dy - (8y + ex ) dx, where C is the boundary of the square with vertices (4, 4), (7, 4), (7, 7), and (4, 7).arrow_forwardUsing Green's theorem, evaluate | C(r) • dr counterclockwise around C the boundary curve C of the region R, where F = [e*+v, e"-, R the triangle with vertices (0,0), (3, 3), (3, 6). NOTE: Enter the exact answer. |C(r) • dr %3Darrow_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_forwardEvaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 5 y²dx + 6 x²dy, where C is the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) oriented counterclockwise. f 5 y²dx + 6x²dy =arrow_forwardChannel flow The flow in a long shallow channel is modeled by the velocity field F = ⟨0, 1 - x2⟩, where R = {(x, y): | x | ≤ 1 and | y | < 5}.a. Sketch R and several streamlines of F.b. Evaluate the curl of F on the lines x = 0, x = 1/4, x = 1/2, and x = 1.c. Compute the circulation on the boundary of the region R.d. How do you explain the fact that the curl of F is nonzero atpoints of R, but the circulation is zero?arrow_forward
- Use Green's Theorem to evaluate · F · dr, where F(x, y) = = with vertices (-3,-9), (5,-9), (5,2), and (-3,2). The integral obtained from from Green's Theorem is J dA where D is the interior of the rectangle. This evaluates to (3xy, y 8 +9) and C is the rectanglearrow_forwardVerify: Green's theorem in the plane for f(2x-y³)dx-xydy, where C is the boundary of the region enclosed by the circles x² + y² = 1 and x² + | x² + y² = باتخیرarrow_forwardUse 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_forward
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