Final answer:
The question involves using the Trapezoidal and Simpson's Rule to approximate the integral of x³ from 0 to 2 and calculating the absolute error of these approximations with N=3 and N=5 for the Trapezoidal method and N=3 points for Simpson's Rule.
Step-by-step explanation:
The given question involves computing the exact value of an integral, Trapezoidal approximation, Simpson's approximation, and the absolute error for both approximations using different values of N. To compare, we need to calculate the exact integral of x³ from 0 to 2, which can be done by evaluating the antiderivative at the upper and lower limits.
For the Trapezoidal approximation, the interval [0, 2] is divided into N equal parts and the area of trapezoids formed by the function given at these intervals are summed. For Simpson's approximation, we also divide the interval into an even number of segments, use parabolic arcs to approximate the function, and sum the areas. N needs to be an even number for Simpson's rule, so we can only apply it with N=3 if we consider N=2 intervals (3 points).
To find the absolute error, subtract the approximation from the exact integral value. The exact value is ∫[0, 2] x³ dx = 24/4 - 04/4 = 4. The Trapezoid approximation and Simpson's approximation will be compared against this value.