Final answer:
The question requires using the Trapezoidal Rule and Simpson's Rule to approximate the value of the integral of x² from 1 to 7, then compare the results with the exact value of the integral.
Step-by-step explanation:
The question asks for the approximation of the definite integral of the function x² from 1 to 7 using the Trapezoidal Rule and Simpson's Rule. To apply the Trapezoidal Rule, one typically divides the interval into 'n' equal subintervals and then uses the formula:
½ * h * [f(x₀) + 2*f(x₁) + 2*f(x₂) + ... + 2*f(xₙ-1) + f(xₙ)],
where h = (b-a)/n and xₙ are the endpoints of the subintervals. Simpson's Rule, on the other hand, also divides the interval into 'n' equal subintervals, but these must be an even number. The formula used is:
⅛ * h * [f(x₀) + 4*f(x₁) + 2*f(x₂) + 4*f(x₃) + ... + 4*f(xₙ-1) + f(xₙ)].
After calculating the approximations using both rules, you should compare them to the exact value of the integral, which can be found by evaluating the antiderivative of x² from 1 to 7. The exact value is given by ⅛ * [x³] evaluated from 1 to 7.