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The function, f(x) = e^-x can be used to generate the following table of unequally spaced data: Evaluate the integral from a=0 b=1.2 using (a) analytical rule, (b) the trapezoidal rule, and (c) a combination of the trapezoidal and Simpson's rules: employ Simpson's rules (1/3, 3/8) wherever possible to obtain the highest accuracy. For (b) and (c) compute the percent relative error.

User Brasofilo
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Final answer:

To evaluate the integral of the function f(x) = e^-x from a=0 to b=1.2, you can use the analytical rule, the trapezoidal rule, or a combination of the trapezoidal and Simpson's rules. The trapezoidal rule divides the interval into small trapezoids and sums their areas, while Simpson's rule provides a more accurate approximation using equally spaced data. By applying Simpson's rule (1/3, 3/8) to each subinterval, you can obtain the highest accuracy.

Step-by-step explanation:

To evaluate the integral of the function f(x) = e^-x from a=0 to b=1.2 using different methods:

(a) Analytical Rule:

The integral of the function can be evaluated analytically using the antiderivative. The antiderivative of e^-x is -e^-x, so the integral from 0 to 1.2 is -e^(-1.2) - (-e^0).

(b) Trapezoidal Rule:

The trapezoidal rule approximates the integral by dividing the interval into small trapezoids and summing their areas. In this case, the interval is divided into two subintervals (0 to 1 and 1 to 1.2). The formula for the trapezoidal rule is [(b-a)/2] * [f(a) + f(b)].

(c) Combination of Trapezoidal and Simpson's Rules:

Simpson's rule is more accurate than the trapezoidal rule, but it requires equally spaced data. However, we can still use Simpson's rule (1/3, 3/8) to improve accuracy. The interval is divided into two subintervals (0 to 1 and 1 to 1.2) using the trapezoidal rule, and then Simpson's rule is applied to each subinterval to obtain the final result.

User Potockan
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