Approximate the Definite Integral

Question

Use series to approximate the definite integral to within the indicated accuracy:

∫ lower bound 0, upper bound 0.4 (e^−x^2)dx, with an error < 10^−4

Note: The answer you derive here should be the partial sum of an appropriate series (the number of terms determined by an error estimate). This number is not necessarily the correct value of the integral truncated to the correct number of decimal places.

Answer 1

The infinite series for e^x = ∑xⁿ / n!.

The infinite series for e^{-x^2} = ∑ (-1)ⁿ(x²ⁿ) / n! , or a_n = (-1)ⁿ(x²ⁿ) / n!

Integrating this make a_n = (-1)ⁿx²ⁿ⁺¹/ (n!(2n+1)) which is an alternating series. For an alternating series, the error is defined: |a_n+1| < error. So we need to find the number of terms, n, where

(0.4)²ⁿ⁺¹/(n!(2n+1)) < 0.0001

a₀ = 0.4¹/(1(1)) = 0.4

a₁ = -0.4³/(1(3)) = -0.0213333

a₂ = 0.4⁵/(2(5)) = 0.001024

a3 = -0.4⁷/(3!(7)) = -0.000039 --> |a₃| < 0.0001

So only need 0.4 - 0.0213333 + 0.001024 = 0.03796907

Note, the actual value = 0.3796528397004753, so a pretty good approximation.