Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the given value of \(n .\) Round your answers to four decimal places and compare your results with the exact value of the definite integral. $$ \int_{0}^{8} \sqrt[3]{x} d x, \quad n=8 $$

Short answer

After performing all the calculations according to the given rules and the exact definite integral, you should have three numerical values. By comparing these, you see how the Trapezoidal Rule and Simpson's Rule perform at approximating the integral in this case.

Step-by-step solution

Calculation using the Trapezoidal rule

First, we compute the approximation using the Trapezoidal Rule. Given the interval [a,b], function f(x), and n partitions, the Trapezoidal Rule follows the formula: \[ T_n = \frac{b - a}{2n} [ f(a) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(b)] \]Substituting the given values, we compute: \[ T_8 = \frac{8 - 0}{2*8} [ f(0) + 2f(1) + 2f(2) + ... + 2f(7) + f(8)] \]Plugging \[f(x) = \sqrt[3]{x}\] into the function arguments, perform the calculation and round the answer to four decimal places.

Calculation using Simpson's rule

Now, compute the approximation using Simpson's Rule. Given the same inputs, the Simpson's Rule follows the formula: \[ S_n = \frac{b - a}{3n} [ f(a) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-2}) +2f(x_{n-1}) + f(b) ] \]Then, implement this formula with the given function and interval: \[ S_8 = \frac{8 - 0}{3*8} [ f(0) + 4f(1) + 2f(2) + ... + 4f(7) + f(8)] \]Substitute \[f(x) = \sqrt[3]{x}\] for the function arguments, calculate and round the result to four decimal places.

Calculation of the exact definite integral

To find the exact value of the integral, we integrate the given function on the interval from 0 to 8. The integral of \[\sqrt[3]{x}\] is \[\frac{3}{4} x^{4/3}\], and using the Fundamental Theorem of Calculus, the definite integral \[ \int_{0}^{8} \sqrt[3]{x} dx \] becomes \[\frac{3}{4} * 8^{4/3} - \frac{3}{4} * 0^{4/3}\]. Perform this calculation and round the result to four decimal places.

Comparison of the results

Finally, look at the results from the Trapezoidal Rule, Simpson's Rule, and the exact value of the definite integral. Observe how close or far the estimates from the rules are to the exact value. The smaller the difference, the more accurate the rule.