To approximate to three decimal places

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

To approximate

\displaystyle \int_{0}^{1} e^{-x^{2}}\,dx

to three decimal places, which of the following approaches is most appropriate? (a) Apply the substitution

u=-x^{2}

to obtain an elementary antiderivative and evaluate directly. (b) Use a single trapezoidal estimate on

[0,1]

; the smoothness of

e^{-x^{2}}

guarantees three-decimal accuracy. (c) Expand

e^{-x^{2}}

into its Maclaurin series and sum enough terms until the remainder is less than

10^{-3}

. (d) Rewrite

e^{-x^{2}}

(e^{-x})^{\,x}

and integrate by treating the exponent as a parameter.