Verifying Green's Theorem In Exercises 5-8, verify Green’s Theorem by evaluating both
for the given path.
C: rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)
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- Evaluate 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_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_forwardFinding u and du In Exercises 1–4, complete the table by identifying u and du for the integral. 1.. | F(9(x))/(x) dx u = g(x) du = g' (x) dx | (52? +1)*(10z) dæ | f(9(2))/(2) dæ 1 = g(x) du = g (x) dx 2 /æ³ +1 dx 3. | Fo(2))/ (x) dz = g(z) du = g (x) dæ tan? x sec? x dx 4. | f(g(x))g(x) dæ u = g(x) du = g (x) dx COs e sin? 2.arrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. ²dx + 2x²dy, where C is the square with vertices (0, 0), (3, 0). (3, 3), and (0, 3) oriented counterclockwise. fy²dx + 2x²dy =arrow_forwardEvaluating a Line Integral Using Green's Theorem In Exercise, use Green's Theorem to evaluate the line integral. √(√(x² - 1²) C: r = 1 + cos 8 (x² - y²) dx + 2xy dyarrow_forwardGreen's Second Identity Prove Green's Second Identity for scalar-valued functions u and v defined on a region D: (uv²v – vv²u) dV = || (uvv – vVu) •n dS. (Hint: Reverse the roles of u and v in Green's First Identity.)arrow_forward
- Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly. $ 5 y dx + 5 x²dy, where Cis the square with vertices (0, 0), (2, 0), (2, 2), and (0, 2) oriented counterclockwise. + iarrow_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 · 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_forward
- сп show that f(x=3x is integralble [0,4] using the definition.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_forwardCrevenie a) let F = (y+z )i +(z+x)j+(y+y)k find a) Curl F b) Divergunce of Farrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,