Compute the error (the difference between the exact value and the approximation) to 5 decimal places in each of the Midpoint, Trapezoidal and Simpson's Rule approximations using .
A) Midpoint error: 0.02033 Trapezoidal error: 0.12109 Simpson's error: 0.08048
B) Midpoint error: 0.00019 Trapezoidal error: 0.12109 Simpson's error: 0.08048
C) Midpoint error: 0.02033 Trapezoidal error: 0.00038 Simpson's error: 0.00000
D) Midpoint error: 0.00019 Trapezoidal error: 0.00038 Simpson's error: 0.00000

Question

Compute the error (the difference between the exact value and the approximation) to 5 decimal places in each of the Midpoint, Trapezoidal and Simpson's Rule approximations using  . 
A) Midpoint error: 0.02033 Trapezoidal error: 0.12109 Simpson's error: 0.08048 
B) Midpoint error: 0.00019 Trapezoidal error: 0.12109 Simpson's error: 0.08048 
C) Midpoint error: 0.02033 Trapezoidal error: 0.00038 Simpson's error: 0.00000 
D) Midpoint error: 0.00019 Trapezoidal error: 0.00038 Simpson's error: 0.00000

Quizplus · Accepted Answer

In this problem, we are given the exact value of the integral and asked to compute the errors for the Midpoint, Trapezoidal, and Simpson's Rule approximations. To find the errors, we need to subtract the exact value from each of the three approximations and take the absolute value.

We can see that the Midpoint and Trapezoidal errors are relatively large compared to the Simpson's error. This is because Simpson's Rule is a higher order method that uses a quadratic polynomial to approximate the curve, while the Midpoint and Trapezoidal methods use linear approximations. Therefore, the Simpson's Rule approximation is expected to be more accurate.

Since the Simpson's error is much smaller than the Midpoint and Trapezoidal errors, we can conclude that Simpson's Rule is the best choice for approximating this integral. Therefore, the answer is D.

Compute the Error (The Difference Between the Exact Value and the Approximation)