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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.

$\int_{0}^{\infty} e^{-x^{2}} d x \approx 0.886196$

Integration Techniques

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