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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 + yUse the transformation u = y - x, v = y, to evaluate the integral on the parallelogram R of vertices (0, 0), (1, 0), (2, 1), and (1, 1) shown in the figure. (y2 - xy) dA 1 0.8 0.6 0.4- 0.2- 0.5 1 1.5QA-Evaluate the double integral [√x + √ay dedy where R is the region in the first quadrant in the xy-plane bounded by the hyperbolas xy-1, x-4 and the lines y 4x and y-9x using the transformation x-u, y-uv with u>0, v>0.
- Find the area of the region bounded by the curvesxy= 1,xy= 3,y= 2x, and y= 3x by using the transformation u=xy and v=yxUse the transformation u = 2x + y, v=x + 3y to evaluate the given integral for the region R 1 1 bounded by the lines y=-2x+3, y=-2x+4, y = -x, and y=-3x+3. (2x² + 7xy+3y²) dx dy R (2x² + 7xy + 3y²) dx dy=[ R (Simplify your answer.)Find div F and curl F if F(x,y,z)=11y6z5i-17x7z10j-6xy7k
- Find div F and curl F of F(x, y,z) = (x z³) i + (3x²y12 )j + (3z²y) k %3DThe centroid of the plane region bounded by the graphs of y = f (x), y = 0, x = 0, and x = 3 is (1.2, 1.4). Without integrating, find the centroid of each of the regions bounded by the graphs of the following sets of equations. Explain your reasoning.(see the graph as attached here). y = f (x) + 2, y = 2, x = 0, and x = 31. Use the transformation u = ", v = xy to find /| x y³ dA over the region R in the first quadrant enclosed by y = x, y = 3x, xy = 1, xy = 4.
- Let 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 theUse the transformation u = 2x + y, v = x + 4y to evaluate the given integral for the region R bounded by the lines 1 y = - 2x + 2, y = - 2x + 4, y = - -x, and y= 4 (2x² +9xy + 4y²) dx dy R (2x² + 9xy + 4y²) dx dy=[ R (Simplify your answer.) 1 --x+1. 4