# Ver2 a Sketch the region D , in R 2 , enclosed by the...

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1ver2. (a) Sketch the region D , in R 2 , enclosed by the curves 2 x + y = 1, 2 x + y = 3, y - 3 x = 0 and y - 3 x = 1. (b) Compute the area of D . 1ver2. (a) Sketch the region D , in R 2 , enclosed by the curves x + 2 y = 0, x + 2 y = 4, y - 3 x = - 1 and y - 3 x = 2. (b) Compute the area of D . 2ver1. Let F ( x, y ) := (sin x, x 2 y 3 ) and C be the triangle in the plane with corner points (0 , 0), (2 , 0) and (2 , 2). Find the counterclockwise circulation of F around C . 2ver1. Let C be the triangle in the plane with corner points (0 , 0), (2 , 0) and (2 , 2). Com- pute C - sin x dx + x 2 y 3 dy . 3ver2. (a) State the formula at the conclusion of Stokes’ Theorem. (b) Evaluate the integral S ( ∇ × F ) · n dS where F ( x, y, z ) := - xz i + yz j + xye z k and S is the cap of the paraboloid z = 5 - x 2 - y 2 above the plane z = 3. Assume n points in the negative z -direction on S . 3ver2. (a) State the formula at the conclusion of Stokes’ Theorem. (b) Evaluate the integral S ( ∇ × F ) · n dS where F ( x, y, z ) := - xz i + yz j + xye z k and S is the cap of the paraboloid z = 8 - x 2 - y 2 above the plane z = 3. Assume n points in the negative z -direction on S . 4ver1. Suppose that y = f ( x ), where f is a differentiable function. The surface, in R 3 , generated by rotating f ( x ) around the y -axis from a x b is parametrized by