Final answer:
The two-segment trapezoidal rule can be used to estimate the integral of a function. We can calculate the values of f(a), f(x1), and f(b), and plug them into the formula to estimate the integral. The error in estimating the integral can be calculated using the composite trapezoidal rule error formula.
Step-by-step explanation:
The two-segment trapezoidal rule can be used to estimate the integral of a function. To use this method, we divide the interval [a, b] into two segments. In this case, the interval is from 0 to 0.8. The formula for the two-segment trapezoidal rule is:
Δx/2 * (f(a) + 2*f(x1) + f(b)), where Δx = (b-a)/2, x1 = a + Δx, and f(x) is the function.
Using the given function f(x) = 0.2+25x−200x²+675x³−900x⁴+400x⁵ and the interval [0, 0.8], we calculate the values of f(a), f(x1), and f(b). Then we plug these values into the formula to estimate the integral.
The error in estimating the integral using the two-segment trapezoidal rule can be calculated using the composite trapezoidal rule error formula:
Error = (b-a)^3 * (f''(c))/12, where c is a value between a and b, and f''(c) is the second derivative of the function.