Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of n. Compare these results with the exact value of the definite integral. Round your answers to four decimal places. $\int _ { 0 } ^ { 1 } \left( \frac { x ^ { 2 } } { 2 } + 1 \right) d x , n = 4$
A)a. Exact: $\approx 1.9667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.1667$
B)a. Exact: $\approx 1.1667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.9667$
C)a. Exact: $\approx 1.1667$ b. Trapezoidal Rule: $\approx 1.9719$ c. Simpson's Rule: $\approx 1.1667$
D)a. Exact: $\approx 2.1667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.1667$
E)a. Exact: $\approx 1.1667$ b. Trapezoidal Rule: $\approx 1.1719$ c. Simpson's Rule: $\approx 1.1667$

Question

Quizplus · Accepted Answer

The exact value of the integral $\int _ { 0 } ^ { 1 } \left( \frac { x ^ { 2 } } { 2 } + 1 \right) d x$ can be found by integrating the function to get $\frac{x^3}{6} + x$ evaluated from 0 to 1, which equals $\frac{1}{6} + 1 = 1.1667$. The Trapezoidal Rule and Simpson's Rule are numerical methods for approximating the value of a definite integral. The Trapezoidal Rule approximation might not exactly match the exact value, but Simpson's Rule, especially with an even number of intervals like $n = 4$, often provides a very accurate approximation for polynomials up to the third degree. In this case, both methods should give approximations close to the exact value, but Simpson's Rule is expected to be very accurate due to the nature of the function being integrated.

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