Let for whole numbers

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Calculus of change and accumulation · Calculus of change and accumulation

Mathematical Methods for Physics, Ch Accumulation: integration and the techniques that make it possible

Problem

Let

I_n = \displaystyle\int_0^{\pi/2}\sin^n\theta\,\mathrm{d}\theta

for whole numbers

n \ge 0

, so that

I_0 = \pi/2

and

I_1 = 1

. ((a)) Evaluate

I_5

and

I_6

exactly. ((b)) Show that the product

nI_nI_{n-1}

has the same value for every

n \ge 1

, and find that value. ((c)) Explain why

I_{n+1} \le I_n \le I_{n-1}

. Combine this with part (b) to show that

\frac{\pi}{2(n+1)} \le I_n^2 \le \frac{\pi}{2n},

and hence give

I_{100}

to three significant figures without evaluating any integral.

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