Question

1) The integral l = Jo sinx dx is important in the design of precise photographic lenses (among other applications). The function fx)hoever, has no elementary antiderivative, so The sir Fundamental Theorem of Calculus cannot be used to calculate the value of the integral. Use the Trapezoidal Rule, the Midpoint Rule and the Simpson's Rule to approximate the numerical value of the integral. Use n 10. 2) The following data shows the velocity vit) recorded by instruments on board a submarine traveling under the polar icecap directly towards the North Pole. Use the Trapezoidal approximation and the Simpson approximation to estimate the distance s = a v(t)dt traveled by the submarine during the 10-h period from t-0 to t-10. t: 01 2 3 4 5 67 8 9 10 h 3) a) Find the Trapezoidal and Midpoint approximations for 2sinx dx. n 8 b) Estimate the errors involved in the approximations in part (a). c) How large should we choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.00001?

Answer 1

Saathi Date Sin X dx 才. Area 02) [l+ 2(, 998334166 x912) t-f '810363233 t 0-94 147 098s 9 995 63 38S +·ー チ 0, gS5 22 66 =[ 094$83 2032] s+ odd 4(f(x) tffxarfOs).tf(x) tf(%) 스ㅡ [fo Arta ㅡ ertn 乍ーぅ。
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x(mid) (x) x) 0.05 0.99958338 0 0.15 0.996254216 0.1 0.998334166 0.2 0.993346654 0.3 0.985067356 0.4 0.973545856 0.5 0.958851077 0.6 0.941070789 0.7 0.920310982 0.8 0.896695114 0.9 0.870363233 1 0.841470985 0.25 0.989615837 0.35 0.979708021 0.45 0.966590076 0.55 0.950340416 0.65 0.931056009 0.75 0.90885168 0.85 0.8838593 0.95 0.856226847 9.462085788 0.946208579 Mid-point Trapezoidal Rule0.945832072 Simpson's le0.946083169