 # 1. (10 pts) Compute (Hint: Sketch the region on the xy-plane and c...

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1. (10 pts) Compute (Hint: Sketch the region on the xy-plane and convert the whole thing into one double integral.) 2. (10 pts) Let D be the region in the first quadrant bounded by x = 0, y = = y=1-1 as in the figure to the right. Use the y=1-1 change of variables (*) y= to compute the integral D x (Hint: The inverse map of (*) is given by U = THE and 3. (10 pts) Compute the line integral ScF.dr, where y C x and C is the portion of the Archimedes' spiral, oriented counterclockwise, parametrized by = (Hint: Split F into the sum of a conservative field and the vortex field.) 5. (10 pts) Use the Green's Theorem to compute the area of the region D in y the first quadrant bounded by x = 0, y = 0, and the curve x2/3 +y2/3= as in the figure to the right. (Note: You might find S cos2 sin2 tsin4t+Chelpful.) = D 6. (10 pts) Let C be the curve of intersection of the upper hemisphere x2 + y2 + z2 = 9, 2 > 0 and the plane Z = y, oriented clockwise when viewed from above. Compute the line integral ScF.dr, where 2 = - 1 (Hint: Replace the path of integration to a simpler one using y an appropriate theorem.) -2 2 8. (10 pts) Let S be the boundary of that part of the solid elliptical cylinder y2 + 4z2 16 with 1 0. Use the Divergence Theorem to compute the flux IIs F dS, where F(x,y,z) = (3xy,3yz,32x). = 9. (10 pts) Let S be the snowman-like closed surface as in the figure to the right, which is oriented with outward-pointing unit normal. Suppose you know that the unit ball B = {(x,y,z) : x2 + y2 + z2 < 1} is entirely contained in the interior of S. Use the Divergence Theorem to compute the flux given by

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