You need to show your FULL and COMPLETE work for each problem. You MUST show all the details and all the steps. YOU MUST SOD PROBLEMS USING METHODS USED IN CLASS OR IN THE SLIDES. (Question 1) [10+10+10 points] Let D be the region that is bounded by the surface $z = x^2 + y^2$ and the plane $z = 4$. a) Find the triple integral $\iiint_D x dV$. b) Find the triple integral $\iiint_D y dV$. c) If possible, find a value of a so that $\iiint_D (2x + ay)dV = 0$.
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The region D is bounded by the surface z = x^2 + y^2 and the plane z = 4. This means that the values of z range from the surface z = x^2 + y^2 to the plane z = 4. Show more…
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Problem 6. [20 pts] (a) [10 pts] Find the volume of the region beneath z = 2x^3+3x^2+4y^2+10 and above the rectangle with vertices (0,0), (2,0), (2,3), (0,3) in the x-y-plane. (b) [10 pts] Find the double integral ∫∫_R(2x^2 + 4y^2 + 3y)dxdy over the area R: {0 < u < 2, 0 < v < 1} with the transformation {x + 2y = 2u, x - 2y = 2v}
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