Evaluating a Double Integral In Exercises 13-20, set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. 13. \iint_R xy \, dA R: rectangle with vertices (0, 0), (0, 5), (3, 5), (3, 0) 14. \iint_R \sin x \sin y \, dA R: rectangle with vertices (-?, 0), (?, 0), (?, ?/2), (-?, ?/2) 15. \iint_R \frac{y}{x^2 + y^2} \, dA R: trapezoid bounded by y = x, y = 2x, x = 1, x = 2
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Step 2: The length of the rectangle is 3 units and the width is 5 units. Therefore, the area of the rectangle is 3 * 5 = 15 square units. Step 3: For the rectangle with vertices (-7, 0), (1, 0), (1, π/2), (-1, 1/2), the area can be found by calculating the Show more…
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11. Use the change of variables formula for double integrals to calculate the following double integrals over the given region D of the plane: (a) ∬ x^2 dxdy D = quarter disk bounded by the positive x axis, the positive y axis and the circle x^2 + y^2 = 9 (you may assume that cos^2 θ = 1/2(1 + cos 2θ); (b) ∬ x / (x^2 + y^2)^{3/2} dxdy D = portion of annulus bounded by the circles x^2 + y^2 = 1 and x^2 + y^2 = 4 in the first quadrant x ≥ 0, y ≥ 0; (c) ∬ y dxdy D = region bounded by the x axis and the portion of the ellipse x^2/4 + y^2/9 = 1 lying in the upper half plane, y ≥ 0. (Note: For regions involving an ellipse whose equation is x^2/a^2 + y^2/b^2 = 1, an appropriate change of variables is x = ar cos θ y = br sin θ, 0 ≤ r < ∞, 0 ≤ θ ≤ 2π).
Madhur L.
Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R. ∬_R xy dA R: rectangle with vertices (0, 0), (0, 3), (5, 3), (5, 0)
Sri K.
13/ Evaluate the surface integral u222fu222f_S f(x,y,z) dS, a) where f(x,y,z) = 2; S is the portion of the cone between the planes z = 0 and z = 2. b) where f(x,y,z) = x - y - z; S is the portion of the plane x + y = 1 in the first octant between z = 0 and z = 2. 14/ Set up, but do not evaluate, an iterated integral equal to the given surface integral by projecting S on a) the xy-plane, b) the yz-plane, and c) the xz-plane for the integral ∬_S xyz dS, where S is the portion of the plane 2x + 3y + 4z = 12 in the first octant. 15/ Evaluate the surface integral ∬_S f(x,y,z) dS where f(x,y,z) = xyz and the surface S is represented by the vector-valued function r(u,v) = (u cos v) i + (u sin v) j + 3u k, 1 < u < 2, 0 < v < 2.
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