For the following exercises, evaluate the line
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- Use Green's Theorem to evaluate the line integral of F = (x6, 3x) around the boundary of the parallelogram in the following figure (note the orientation). (xo.) (X0.0) Sex6 dx + 3x dy = (2x-Y) ·x With xo = 7 and yo = 7.arrow_forwardCalculate the line integral f(3xy³ − 4x + 4y + 8) dx + (−3xy+5) dy, where C' is the rectangle with vertices (2, −4), (2, -3), (−3,−3), and (−3,-4) oriented clockwise. Enter an exact answer. Provide your answer below: f(3x³ 4x + 4y + 8) dx + (−3xy + 5) dy =arrow_forwardUse the Green's Theorem to evaluate the line integral f, FdT where F = (sin(x²)–2x²y) i´+ (cos(y²) + x³) i and L is a closed curve that consists of a part of the parabola y = x² and the line y = 1, with –1 < x <1 oriented counterclockwise. 6.arrow_forward
- Use Green's Theorem to evaluate the following line integral. Assume the curve is oriented counterclockwise. A sketch is helpful. $(4y - 3,2x² + 8). dr. where C is the boundary of the rectangle with vertices (0,0), (6,0). (6,5), and (0,5) C $(4y - 3,2x² + 8). dr = (Type an exact answer.) Carrow_forwardLet C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (−4, 2), (−4, −3), (2, −2), (2, 7), and back to (-4, 2), in that order. Use Green's theorem to evaluate the following integral. Jo (2xy) dx + (xy2) dyarrow_forwardLet C be the closed, piecewise smooth curve formed by traveling in straight lines between the points (-3, 1), (-3, -2), (2, –1), (2, 4), and back to (-3, 1), in that order. Use Green's theorem to evaluate the following integral. (2xy) dx + (xy²) dyarrow_forward
- |-4 Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. I. $. (x – y) dx + (x + y) dy, C is the circle with center the origin and radius 2 -arrow_forwardConsider the line integral xy dx + x²y° dy with C the triangle with vertices (0, 0). (1,0), and (1, 2). Evaluate this line integral by two methods: (a) directly, and (b) by using Green's Theorem. Orient C in the counterclockwise direction. (,?) (1,0) Hint: the curve C'is composed of three line segments, so you will have to compute the line integral along each segment separately.arrow_forwardI need help with #5arrow_forward
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