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

Question

Quizplus · Accepted Answer

Both the trapezoidal rule and Simpson's rule require an even number of subintervals, which is not the case for options A and C. Option D has a typo and the upper limit of integration should be 3 instead of 1. For option B, using 4 subintervals, we get:

Trapezoidal rule:
$\Delta x = \frac{3-1}{4} = 0.5$
$$\int_{1}^{3} \frac{1}{x} dx \approx \frac{0.5}{2}\left(\frac{1}{1}+\frac{1}{1.5}+\frac{1}{2}+\frac{1}{2.5}+\frac{1}{3}ight) \approx 0.9867$$

Simpson's rule:
$\Delta x = \frac{3-1}{4	imes 2} = 0.25$
$$\int_{1}^{3} \frac{1}{x} dx \approx \frac{0.25}{3}\left(\frac{1}{1}+4\left(\frac{1}{1.5}+\frac{1}{2.5}ight)+\frac{1}{3}ight) \approx 0.9864$$

Option B is the closest to the exact value of the integral, which is $0.9851$.

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