Evaluating a Double Integral by Converting from Rectangular Coordinates [[ (x + y) dA where R = {(x, y)|1 ≤ x² + y² ≤ 4, x ≤ 0} . R Evaluate the integral
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- Instruction: Evaluate the following line integrals in the complex plane by direct integration (not using theorems)· Using an explicit parameterization, perform the following complex con- tour integrals: z" dz [n e Z], $ z-1 dz, sin z dz, where the contours are C1 C2 C3 a b1 aUse 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.
- Q\ Evaluate the iterated integral by converting to polar coordinates V2x-x2 Vx2 + y2 dy dxa) Evaluate the integrals using appropriate substitutions. dx Sino i) S de u) I Cos?8+ 1 ü) j 1+16x2 b) Evaluate the integrals using Part 1 of the Fundamental Theorem of Calculus. /2(x+ 2 L 4x(1 – x?)dx ii) (x + -) dx Sin2x i) c) Use Part 2 of the Fundamental Theorem of Calculus to find the derivatives. evE dt d d i) ii) Intdt dx dxThe area enclosed between the straight line y = x and the parabola y = x2 in the x - y plane is
- 5) Using Green's theorem, convert the line integral f.(6y? 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)Use Green's Theorem to evaluate F · dr, where F(, y) = ( – 2xy, y + 3) and C is the rectangle with vertices (4,-1), (7,-1), (7,9), and (4,9). The integral obtained from from Green's Theorem is SI, dA where D is the interior of the rectangle. This evaluates toApply Green's Theorem to evaluate the integral. + x)dx + (y + 9x)dy C: The circle (x - 5)2 + (y – 9)2 = 9 (Зу + х)dx + (у + 9х)dy %3D (Type an exact answer, using t as needed.)
- (b) Evaluate the line integral Jo dzalong the simple closed contour C shown in the diagram. -2 -1 2j o 1 2Let F = 4(x + y) i + 6 sin(y) j. Find the line integral of F around the perimeter of the rectangle with corners (5, 0), (5, 6), (–2, 6), (-2,0), traversed in that order. line integralWhat equation do I use for an integral of a rotating triangle?