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- Let B be the region in the first quadrant of the xy-plane bounded by the lines x + y = 1, x + y = 2, (x – y)? 0 and y 0. Evaluate - dxdy by applying the transformation u = x + y, v = x – y 1+x + yarrow_forward[(x+ y)e*-° dA Q#2. Set up a transformation and hence evaluate R over the rectangular region enclosed by the lines *+y= 3, x-y = 4, x+ y = 5, x– y= 6. (Sketch the regions are mandatory)arrow_forwardLet B be the region in the first quadrant of the xy-plane bounded by the lines r+ y = 1, x + y = 2, |(x – y)² 1+x + y x = 0 and y = 0. Evaluate -dædy by applying the transformation u = x + y, v = x – y Barrow_forward
- Apply the transformation u = x + y and v = r-y to evaluate I- y dA over the region R enclosed by the lines y = x, y = 1 – 1, y = 1– z and y = 3 – r.arrow_forwardA region R in the xy-plane is given. Find equations for a transformation T that maps a rectangular region S in the uv-plane onto R, where the sides of S are parallel to the u- and v- axes.arrow_forwardIntegrate (x + y) dA, where R = {(x, y):0 < x < 2, x < y < x + 4}. Use the transformation x = 2u and y = 4v + 2u. Hint: Sketch the given region R and then the new region in the uv plane.arrow_forward
- Evaluate the circulation of G = xyi + zj + 4yk around a square of side 4, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis. Circulation = Jo F. dr =arrow_forwardLet B be the region in the first quadrant of the ry-plane bounded by the lines r+y = 1, r + y = 2, (x – y)? I = 0 and y = 0. Evaluate -drdy by applying the transformation u = r+ y, v = x – y 1+x + y Barrow_forwardLet D be the triangular region in the uv-plane with vertices (0, 1), (4, 1), (1, 3) and letG(u, v) = (u − v, 2v).Sketch D in the uv-plane and G(D) in the xy-plane.Find the area of G(D) by using Jac(G)arrow_forward
- Evaluate exp{}dA where R is the region in the ry-plane bounded by the trapezoid with vertices (0, 1), (0, 2), (2,0), and (1,0) by a suitable change of variables.arrow_forwardLet R be the region in the first quadrant bounded by the lines y - xy = 2. hyperbolas xy = 1 2D (a) Consider the transformation T(u, v) = (x,y) given by x = region S in the uv-plane mapped by Tonto R. (b) Evaluate the integral SSR AxydA. and y = 2x and the and yv. Describe thearrow_forward
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