Apply the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral using 8 subintervals. Round your answer to six decimal places and compare the result with the exact value of the definite integral.
$\int _ { 4 } ^ { 5 } \sqrt { 4 + t ^ { 5 } } d t$
A) The Trapezoidal rule gives $43.377044$ and Simpson's rule gives $45.513198$.
B) The Trapezoidal rule gives $43.345903$ and Simpson's rule gives $43.335523$.
C) The Trapezoidal rule gives 43.377044 and Simpson's rule gives 43.335523.
D) The Trapezoidal rule gives $43.345903$ and Simpson's rule gives $45.513198$.
E) The Trapezoidal rule gives $- 43.345903$ and Simpson's rule gives $- 43.335523$.

Question

Quizplus · Accepted Answer

The Trapezoidal and Simpson's rules are numerical methods for approximating definite integrals. The Trapezoidal rule tends to underestimate or overestimate depending on the function's curvature, while Simpson's rule, which uses parabolic arcs instead of straight lines, often provides a more accurate approximation for smooth functions. Given the complexity of the integrand $\sqrt{4 + t^5}$, exact values are difficult to compute without numerical methods, and the discrepancy between the two methods' results reflects their inherent approximation errors. The negative values in option E are incorrect because the integral of a positive function over a positive interval cannot be negative.

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