A Course in Interpolation And Numerical Integration for the Mathematical Laborat

Cover A Course in Interpolation And Numerical Integration for the Mathematical Laborat
A Course in Interpolation And Numerical Integration for the Mathematical Laborat
David Gibb
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3. 5, .. ^ 1 23] NUMERICAL INTEGRATION 65 IT r Jo , . „ ^, ^ 1. 3. 5... {2n-\) - But I siir" ed6= 2.^. % ... -211 ~2 Hence fA')^^i.^<*)=*=. F^y^'W^T^"-.. } If k is small this series is very suitable for obtaining the value of F ; but it is evident that if k approaches unity a large number of terms mu. St be included to get a fair approxima- tion to tlie value of F. Let yl-=;0-I. Then to obtain F correct to 4 decimal places it is only neces- sary to include two terms of the series. For / ^\ ^ \ 0-25 014062-) ) =- ^ { 1 + 0025 + 0-0000 140625 + . . . } = :j {1-0025} = 1-5747. Example 5. — Find the value of /' (^Ic, l) when ^• = sin 10° IT Example 5. — Show that I J \ - k''^ sin- x is equal to 1-54415 when Jc has the value sin 15°. 2 fO-6 Example 4. — Show that the value of the integral — 7=^ I e-'^dx is V TT Jo 0-60386. 24. A Formula of Integration Based on Newton's Formula We shall now show how integration may be performed by the aid of interpolation-formulae. Integrating Newton's interpolation formula 66 INTERPOLATION AND NUMERICAL INTEGRATION [ch.

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