Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral.Find the exact value of the integral.

Question

Quizplus · Accepted Answer

Using the trapezoidal rule with 5 subintervals, we get:

$\Delta x = \frac{3-0}{5} = 0.6$

$T_5 = \frac{0.6}{2} [f(0) + 2f(0.6) + 2f(1.2) + 2f(1.8) + f(3)] = 6.068$

Using Simpson's rule with 2 subintervals, we get:

$\Delta x = \frac{3-0}{2(2)} = 0.75$

$S_2 = \frac{0.75}{3} [f(0) + 4f(0.75) + 2f(1.5) + 4f(2.25) + f(3)] = 6.029$

The exact value of the integral is:

$\int_0^3 \sqrt[3]{x^2} dx = \frac{3}{5} x^{\frac{5}{3}} \biggr |_{0}^{3} = \frac{3}{5} (3^{\frac{5}{3}} - 0) = 6.040$

Therefore, the best approximation is obtained using Simpson's rule with 2 subintervals, which is closest to the exact answer. The only choice that includes Simpson's rule with 2 subintervals is D.

Use the Trapezoidal Rule and Simpson's Rule to Approximate the Value