Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. 4 0 x3 sin(x) dx, n = 8

Step-by-step explanation:

The interval from 0 to 4 is divided into 8 equal parts, so each has a width of 0.5 units. For the trapezoidal and Simpson's rules, the function is evaluated at each end of each interval, and those results are combined in the manner specified by the rule.

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For the trapezoidal rule, the function values are taken as the "bases" of trapezoids, whose "height" is the interval width. The estimate of the integral is the sum of the areas of these trapezoids.

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For the midpoint rule, the function is evaluated at the midpoint of each interval, and that value is multiplied by the interval width to form an estimate of the integral over the interval. In the spreadsheet, midpoints and their function values are listed separately from those used for the other rules. The midpoint area is the rectangle area described here.

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For Simpson's rule, the function values at the ends of each interval are combined with weights of 1, 2, or 4 in a particular pattern. The sum of products is multiplied by 1/3 the interval width. In the spreadsheet, the weights are listed so the SUMPRODUCT function could be used to create the desired total.

We note the Simpson's rule estimate of the integral (-5.61) is very close, as the actual value rounds to -5.64.

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A graph of the function and a computation of the integral is shown in the second attachment.

answer is 67%

step-by-step explanation: