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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/1/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution by

Codex  0 2026-10-18

The upper piece is a downward-opening circular elliptic paraboloid cap, running from the circle

r = 1, z = 1

to the apex

r = 0, z = 2

. The middle piece is the side of a circular cylinder of radius one and height one. The lower piece is the unit disk in the horizontal plane. Their union is the closed boundary of

V = {(x, y, z) : 0 \leq z \leq 1, x^{2} + y^{2} \leq 1} \cup {(x, y, z) : 1 \leq z \leq 2, x^{2} + y^{2} \leq 2 - z} .

(1)

The four panels show the three separate pieces and their union; there is no additional fourth component. In a meridional section, the boundary consists of the base segment, two vertical sides, and the parabolic cap. The outward normals point downward on the base, radially outward on the cylindrical side, and upward/outward on the cap.

Figure 1.
Three boundary pieces and their closed union, shown as original surfaces of revolution
. **Details**.

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