Trapezoidal sums are a method used to approximate the definite integral of a function. Instead of using rectangles like in Riemann sums, we use trapezoids. This often gives a more accurate approximation, especially for functions with curves. The basic idea is to divide the area under the curve into trapezoids and then sum their areas to estimate the total area under the curve, which represents the definite integral. The formula for the area of a single trapezoid is $ \frac{1}{2} (b_1 + b_2)h $, where $b_1$ and $b_2$ are the lengths of the parallel sides (the function values at the endpoints of the interval) and $h$ is the width of the interval.
To calculate the trapezoidal sum over an interval $[a, b]$ divided into $n$ subintervals, we use the formula:
$ T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)] $, where $ \Delta x = \frac{b-a}{n} $ and $ x_i = a + i\Delta x $.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Definite Integral | A. The width of each subinterval |
| 2. Trapezoid | B. A quadrilateral with at least one pair of parallel sides |
| 3. $\Delta x$ | C. The exact area under a curve between two points |
| 4. $f(x_i)$ | D. The value of the function at a specific point $x_i$ |
| 5. $T_n$ | E. The trapezoidal sum approximation with n trapezoids |
Matching Answers: 1-C, 2-B, 3-A, 4-D, 5-E
The trapezoidal rule approximates the __________ of a function by dividing the area under the curve into __________. Unlike Riemann sums which use __________, this method uses __________ to provide a more accurate estimate. The formula requires calculating the __________ of the function at several points within the interval.
Answer: integral, trapezoids, rectangles, trapezoids, value
Explain why the trapezoidal rule generally provides a better approximation than Riemann sums for functions with significant curvature. Provide an example of a function where the trapezoidal rule would give a very accurate result, and another where it might not be as accurate.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀