Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral.
B)
C)
D)

Question

Use the trapezoidal rule and Simpson's rule to approximate the value of the definite integral. ​  ​
B) ​ 
C) ​ 
D) ​

Quizplus · Accepted Answer

Using the trapezoidal rule, we get:
$$\int_{3}^{7}f(x) \, dx \approx \frac{7-3}{2}\left(f(3)+f(7)\right) = 4(f(3)+f(7))$$
$$\int_{3}^{7}f(x) \, dx \approx 4\left(\frac{4}{13}+\frac{4}{49}\right) = \frac{176}{49}$$

Using Simpson's rule, we get:
$$\int_{3}^{7}f(x) \, dx \approx \frac{7-3}{6}\left(f(3)+4f\left(\frac{3+7}{2}\right)+f(7)\right)$$
$$\int_{3}^{7}f(x) \, dx \approx \frac{1}{3}\left(\frac{4}{13}+4\cdot\frac{56}{425}+\frac{4}{49}\right) = \frac{416}{147}$$

Comparing the approximations, we see that $\frac{416}{147}$ is closest to $2.83$, which is also the correct value of the integral. Simpson's rule gives a better approximation than the trapezoidal rule, so the best choice is D, which corresponds to the Simpson's rule approximation.

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