Question

Evaluate the integral x(e²ˣ))x from 0 to 3 with

a) Romberg formula by first applying the composite trapezoid rule with 1, 2, 4, and 8 subintervals to approximate the integral.

Answer 1

To evaluate the integral using the Romberg formula and the composite trapezoid rule, approximate the integral using different numbers of subintervals.

Step-by-step explanation:

To evaluate the integral ∫ x(e²ˣ)dx from 0 to 3 using the Romberg formula and the composite trapezoid rule, we need to first approximate the integral using different numbers of subintervals.

For the composite trapezoid rule with 1 subinterval, we have:

h = (b - a)/n = (3 - 0)/1 = 3

x₀ = 0, x₁ = 3

f(x₀) = x₀(e²ˣ) = 0(1) = 0

f(x₁) = x₁(e²ˣ) = 3(e²³) = 15.08

Approximation = (h/2)(f(x₀) + f(x₁)) = (3/2)(0 + 15.08) = 22.62

Repeat the above steps for 2, 4, and 8 subintervals to find the approximations for each case.