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Module MA2E02 (Frolov), Multivariable Calculus Tutorial Sheet 4 Due: at the end of the tutorial session Tuesday/Thursday, 16/18 February 2016 Name and student number: 1. Use a double integral to find the volume under the surface z = x2 + 2πx3 cos y and over the rectangle R = {(x, y) : 1 ≤ x ≤ 4 , − π2 ≤ y ≤ π2 }. 2. Consider the solid in the first octant bounded by the surface z = 12x3 + 14x2 y + 8xy 2 , below by the plane z = 0, and laterally by y = x2 and y = x. (a) Sketch the projection of the solid onto the xy-plane. (b) Use double integration to find the volume of the solid. Show the details of your work. 3. Consider the solid inside the surface r2 + z 2 = 16 and outside the surface r = 3. Here r2 = x2 + y 2 . (a) What is the surface r2 + z 2 = 16? (b) What is the surface r = 3? (c) Sketch the projection of the solid onto the xy-plane. (d) Use double integration and polar coordinates to find the volume of the solid. Show the details of your work. 1