Question

Evaluate the integral I = ∫[0 to 3] x * e^(2x) dx with a) Romberg formula by first applying the composite trapezoid rule with 1, 2, 4, and 8 s

Answer 1

To evaluate the integral, you can use the composite trapezoid rule by dividing the interval into subintervals and applying the trapezoidal rule to each subinterval. The formula for the composite trapezoid rule helps approximate the integral by summing up the approximations over all the subintervals.

Step-by-step explanation:

To evaluate the integral ∫[0 to 3] x * e^(2x) dx using the Romberg formula, we first apply the composite trapezoid rule with 1, 2, 4, and 8 subintervals. The composite trapezoid rule approximates the integral by dividing the interval [0, 3] into smaller subintervals and using the trapezoidal rule to approximate the integral over each subinterval. We then sum up the approximations over all the subintervals to obtain an approximation for the integral.

The formula for the composite trapezoid rule is:

I ≈ h/2 * (f(a) + 2*sum(f(xi)) + f(b))

where h is the width of each subinterval, a is the lower limit of integration, b is the upper limit of integration, and xi represents the points within each subinterval.