Final answer:
To approximate the integral of (5-x^2) from 1 to 4 with n=6, both the Trapezoidal Rule and Simpson's Rule divide the interval into equal parts and use those subdivisions to estimate the area under the curve. Results are then rounded to four decimal places and compared to the exact value of the integral.
Step-by-step explanation:
To approximate the value of the definite integral ∫ 14 (5−x2)dx with n=6 using the Trapezoidal Rule and Simpson's Rule, first we must set up both rules for the given function and interval.
Trapezoidal Rule
The Trapezoidal Rule approximates the integral by summing the areas of trapezoids under the curve. The interval from 1 to 4 is divided into n=6 equal parts, giving us a delta x=(4-1)/6. The endpoints and midpoints are then plugged into the function f(x)=5−x2 and summed with proper coefficients.
Simpson's Rule
Simpson's Rule approximates the integral by using parabolic arcs rather than straight lines to approximate the curve. The interval is similarly subdivided, but the summation includes coefficients related to whether the term is an endpoint, an odd midpoint, or an even midpoint. This requires an even number of intervals, which we have (n=6).
Finally, each method's result is calculated and rounded to four decimal places before being compared to the exact value of the integral. The exact value can be found by directly integrating the function from 1 to 4.