Question

Approximate the definite integral using the Trapezoidal Rule and Simpson's Rule. Compare these results with the approximation of the integral using a graphing utility. (Round your answers to four

decimal places.)
∫₀¹√(x)√(1-x) d x₁ n=4

Answer 1

To approximate the given integral from 0 to 1 using the Trapezoidal Rule and Simpson's Rule, divide the interval into 4 equal parts, calculate the function values at specific points, and apply the respective formula for each rule. Comparing these results to a graphing utility's numerical integration will reveal their accuracy and efficiency.

Step-by-step explanation:

To approximate the definite integral of the function √(x)√(1-x) from 0 to 1 using the Trapezoidal Rule and Simpson's Rule, first we need to split the interval [0, 1] into n equal parts, where n is given as 4. Using the Trapezoidal Rule, we evaluate the function at the endpoints and at the midpoints, then apply the formula:

• h = (b - a) / n
• Trapezoidal sum = (h/2) * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)]

For Simpson's Rule, the formula is slightly different:

• Simpson's sum = (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4)]

Calculate each term using the function value at the given x-points and then calculate the final approximation by plugging these values into the respective formulas.

For comparison, you can use a graphing utility to numerically integrate the function and round the result to four decimal places. The graphing utility should provide a result, which will allow you to compare the efficiency and accuracy of the Trapezoidal Rule and Simpson's Rule.