 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/12/d/solution

For

a /∣ b ∣ ≪ 1

, apply the binomial series to the result in (c):

(1 + \frac{4 a ^{2}}{b ^{2}})^{3/2} = 1 + \frac{6 a ^{2}}{b ^{2}} + \frac{6 a ^{4}}{b ^{4}} + O (a^{6} / b^{6}) .

Therefore

A = \frac{α b ^{2}}{2} + \frac{α a ^{2}}{2} + O (a^{4} / b^{2}) \sim \frac{α b ^{2}}{2} .

For

α = 2 π

, the leading term is

π b^{2}

, the area of the base disk. A very shallow cap is nearly planar; its first area excess is

π a^{2}

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