numerical integrationStewart Calculus 8e. 7.7. selected homework exercises worked.2. linear (concave up decreasing function)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So 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 for a concave up decreasing function.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkobXN1YnN1cEdGJDYnLUkjbW9HRiQ2LVErJkludGVncmFsO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmdW5zZXRGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZHLUYjNictSSNtbkdGJDYkUSIwRidGMi8lJ2l0YWxpY0dGPC8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjwvRjNRJ2l0YWxpY0YnLUYjNictRk02JFEiMkYnRjJGUEZSRlVGVy8lMXN1cGVyc2NyaXB0c2hpZnRHRk8vJS9zdWJzY3JpcHRzaGlmdEdGTy1JJW1zdXBHRiQ2JS1GLzYtUS8mRXhwb25lbnRpYWxFO0YnRjJGNUY4L0Y7RjdGPS9GQEY3RkFGQ0ZFL0ZJUSwwLjExMTExMTFlbUYnLUYjNiktRi82LVEqJnVtaW51czA7RidGMi9GNlEmZmFsc2VGJy9GOUZccC9GO0ZccC9GPkZccC9GQEZccC9GQkZccC9GREZccC9GRlEsMC4yMjIyMjIyZW1GJy9GSUZkcC1JI21pR0YkNiVRInhGJ0ZQRldGUC9GU1ErWzAsMTYwLDgwXUYnRlUvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjxGV0Zobi1GZ3A2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuNWVtRicvJSZkZXB0aEdGZnEvJSpsaW5lYnJlYWtHUSVhdXRvRictRi82LVEwJkRpZmZlcmVudGlhbEQ7RidGMkY1RjhGYm9GPUZjb0ZBRkNGRUZIRmZwLUYvNi1RIjtGJ0YyRltwL0Y5RjxGXnBGX3BGYHBGYXBGYnBGRS9GSVEsMC4yNzc3Nzc4ZW1GJy1GZ3A2JVEmZXZhbGZGJ0ZQRlctSShtZmVuY2VkR0YkNiQtRiM2JS1GZ3A2JVEiJUYnRlBGVy8lK2V4ZWN1dGFibGVHRlxwRjJGMkZjc0Yycomparison of the various approximations explicitlyWe can compare different approaches to approximating this integral, first with the Tutor. This is nearly identical to the following 10 division function area calculation.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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjxGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJNRidGL0YyRjUtRiw2JVEmRXhhY3RGJ0YvRjJGNS1GLDYlUSJURidGL0YyRjUtRiw2JVEiTEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOQ== for a concave up decreasing function, so we order them this way: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JSFH3. linear (concave down decreasing function)The concave down decreasing function interchanges the left and right endpoint approximations in the previous ordering.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUTRBcHByb3hpbWF0ZUludFR1dG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiZkYnRjRGNy1GOzYkLUYjNiQtRiw2JVEieEYnRjRGNy9GOFEnbm9ybWFsRidGSUZJRklGSUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRidGSQ==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So for a concave down decreasing function, one should have (reverse order from the concave up decreasing case):
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. simpson (concave down increasing function)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInhGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYjNiUtRiw2I1EhRictSSZtc3FydEdGJDYjLUYjNiYtSSVtc3VwR0YkNiVGTy1GIzYlLUkjbW5HRiQ2JFEiM0YnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORlxwLUZibzYkUSIxRidGOUY5RjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSEZVRk0tRiw2JVElcGxvdEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYzRistRmlwNiQtRiM2JEZPRjlGOS1GNjYtUSIsRidGOUY7RmRwRkBGQkZERkZGSEZVL0ZOUSwwLjMzMzMzMzNlbUYnRk8tRjY2LVEiPUYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GXnAtRjY2LVEjLi5GJ0Y5RjtGPkZARkJGREZGRkhGW3BGVy1GYm82JFEiMkYnRjlGYXEtRiw2JVEnZmlsbGVkRidGL0YyRmZxLUZpcDYmLUYjNiYtRiw2JVEmY29sb3JGJ0YvRjJGZnEtRiw2JVElcGlua0YnRi9GMkY5RjkvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGYXEtRiw2JVEqZ3JpZGxpbmVzRidGL0YyRmZxLUYsNiVGMUYvRjJGOUY5RmFwLUYsNiVRJXBsdDBGJ0YvRjJGNS1GLDYlUSIlRidGL0YyLUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGSkZNLyUrZXhlY3V0YWJsZUdGPUY5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVE0QXBwcm94aW1hdGVJbnRUdXRvckYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiNRIUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkBGPUZAFor a concave down increasing function only LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn are interchanged in the concave down decreasing situation:
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 for a concave down decreasing function, one should have (reverse order from the concave up decreasing case), and we added in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRjIvRjZRJ25vcm1hbEYn which is closest to the exact result but a bit under, no way to see this from the geometry.
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30. tabular data, simpsonJust edit the (odd number) data points, here 9 for 8 divisions, and edit the Simpson coefficients for a different odd number: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JSFH35. graphical data, simpsonWe cannot estimate the function values to better than about plus or minus 1/4 for the net change in the velocity from integrating the acceleration from this 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are my estimates for the 7 endpoint values of the 6 second interval: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so how can we estimate the error bar here? Certainly the first decimal digit is not trustable, but how many tenths above and below should we guess? Our estimates of the function values at unit intervals may differ in a window of plus or minus 0.25, but randomly some guesses will be above, some below, so ...I think we need a statistician.42. function expression, 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that for this integral, 10 division Simpson rule (plust 10 digit approximation for the angle) actually gives the 10 digit result modulo that final digit. But we cannot trust the last digit by adding the 11 values at 10 digit accuracy. We would need to carry out the calculation with more digits to trust that last digit.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 the Maple command seems to nail the 10th digit if we use more digits. Our naive sum of values relies on a 10 digit angle in intermediate calculations, but still gives the right 10th digit. Hmm. No matter, the takeaway is the fact that only 10 divisions nearly guarantee 10 digit accuracy in this example.