Use the properties of integrals to verify that
2 <= ∫ -1 to 1 (1 + x^2) ^ 1/2 dx <= 2(2) ^ 1/2
Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows that
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