Homework SourseUse series to approximate the definite integral to within th
Use series to approximate the definite integral to within the indicated accuracy:
the integral from from 0to 0.4 of e^?x^3 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.
please show work so I can understand. Thank you!
Solution
e^x= 1+x+x^2/2!+x^3/3! +x^4/4! sol start with e^(-x^3)=1-x^3+(-x^3)^2/2!+(-x^3)^3/3!… then we have e^(-x^3)=1-x^3-x^6/2!-x^9/3!-x^12/4! integrate each term INT[e^(-x^3) dx]=x-(1/4)x^4 - x^7/(14) -x^9/(48) -x^13/(312) so plug in 0.4 Basically which ever term is the first to be less than 0.0001 is going to be left out and that will be your error. so you might have to add more terms. ans
