Use Simpson's rule with \( n=4 \) to approximate the integral \( \int_{0}^{1} e^{x^{2}} d x \)
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The function is \( f(x) = e^{x^2} \) and the interval is \([0, 1]\). Show more…
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Estimate the numerical value of $ \displaystyle \int_0^\infty e^{-x^2}\ dx $ by writing it as the sum of $ \displaystyle \int_0^4 e^{-x^2}\ dx $ and $ \displaystyle \int_4^\infty e^{-x^2}\ dx $. Approximate the first integral by using Simpson's Rule with $ n = 8 $ and show that the second integral is smaller than $ \displaystyle \int_4^\infty e^{-4x}\ dx $, which is less than 0.0000001.
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Calculate $S_{N}$ given by Simpson's Rule for the value of $N$ indicated. $$ \int_{0}^{1} e^{-x^{2}} d x, \quad N=4 $$
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