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.