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Problem 60 Hard Difficulty

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

$ \int^{0.5}_0 x^2 e^{-x^{2}} dx $ $ \left( \mid \text {error} \mid < 0.001 \right) $

Answer

0.03653

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Video Transcript

this problem. We want to find the series representation of X Squared E to the negative X squared, using the series for E to the X. So the series E to the X is modeled by the some summation from an equal zero to infinity of X to the end over and Factorial, which is equal to one plus X over one factorial plus x two squared over two factorial plus X cubed over three factorial and so on. And the radius of convergence is infinity now X to the negative or each of the negative. X squared, however, is now going to look like the some from an equal zero to infinity. Still negative one to the end, though now, because there's a negative exponents and instead of X, the nlb X to the two n over and factorial then lastly, we have X squared E to the negative X squared, which is going to end up equaling the some from an equal 02 infinity of negative one to the end of X to the two n plus two over and factorial. So we see that as we progress, it looks more and more like this final form then if we integrate using the series, um, what will end up getting when we integrate this series? DX is we end up getting the, uh, some from an equal zero to infinity of negative one to the end. Time 0.5 to 1 plus three over two and plus three and factorial serum. And then for the alternating series, we want to look at the term a N That is less than the required accuracy. So a N is going to be less than 0.1 Um, so when we use this value, we see that will only need to use two terms. So when we do our two terms and calculate it, we will end up getting approximately 0.35 as our value. Um, And if we do more terms, we see that it would get closer to 0.36