Final answer:
a. The Trapezoidal Rule is a numerical method used to approximate the area under a curve. b. Riemann Sums divide the interval into smaller subintervals to estimate the area. c. The Trapezoidal Rule is commonly used in calculus to approximate definite integrals. d. Simpson's Rule is another method for numerical integration that uses parabolic curves.
Step-by-step explanation:
a) The Trapezoidal Rule serves as a numerical technique for estimating the area beneath a curve.
It involves partitioning the area into trapezoids and summing their individual areas to provide an approximation of the overall area.
b) When employing Riemann Sums to ascertain the area beneath a curve, the interval is subdivided into smaller segments, and the area of each subsection is approximated using rectangles.
The cumulative sum of these areas then yields an estimation of the total area.
c) In calculus, the Trapezoidal Rule finds application in approximating definite integrals of functions, particularly when the analytical integration of the function is challenging or impractical.
d) Comparing the Trapezoidal Rule and Simpson's Rule in numerical integration, both methods aim to approximate the area under a curve.
While the Trapezoidal Rule utilizes trapezoids for approximation, Simpson's Rule employs parabolic curves.
Generally, Simpson's Rule yields more accurate results than the Trapezoidal Rule, albeit at the expense of requiring more computational complexity.