Riley hobson bence · Problem 2.46

Preliminary calculus

Mathematics — Methods · integral-bounds-substitution

Problem

Find positive constants

a

b

such that

ax \leq \sin x \leq bx

for

0 \leq x \leq \pi/2

. Use this inequality to find (to two significant figures) upper and lower bounds for the integral

I = \int_0^{\pi/2} (1 + \sin x)^{1/2}\, dx.

Use the substitution

t = \tan(x/2)

to evaluate

I

exactly.