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.
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