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Chapter 7: Problem 3

Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.

Short Answer

Expert verified
Explain geometrically how the Trapezoid Rule is used to approximate a definite integral. The Trapezoid Rule is a method used to approximate the definite integral, which represents the signed area under a curve between two points on the x-axis. This geometric technique involves dividing the area under the curve into a series of trapezoids and summing their areas. By constructing trapezoids using the function's graph and the x-axis as the base, the shorter and longer parallel sides of each trapezoid are determined by the function's values at the endpoints of the subintervals. The trapezoids' areas are then computed and summed to approximate the definite integral. As more trapezoids (higher value of n) are used in the approximation, the result converges to the actual value of the definite integral, providing a geometric interpretation of the integration process.

Step by step solution

01

Define a Definite Integral and The Trapezoid Rule

A definite integral is a mathematical concept that represents the signed area under a curve, between two points on the x-axis. This is represented as 鈭玣(x) dx from a to b for a function f(x). The Trapezoid Rule is a technique for approximating this definite integral by dividing the area under the curve into a series of trapezoids and summing their areas.
02

Subdivide the Area Under the Curve

Begin by selecting a fixed number of equally spaced points along the x-axis, within the interval [a, b]. Starting from the leftmost point, label these points as x鈧�, x鈧�, x鈧�, 鈥�, x鈧�, where x鈧� = a and x鈧� = b. The number of divisions, n, determines the number of trapezoids used to approximate the area under the curve.
03

Construct Trapezoids

Create a trapezoid from the function's graph at each subdivision, using the x-axis as the base. For each trapezoid, the shorter parallel side (vertical side) is determined by the function's value at the left subinterval endpoint (f(x岬⑩倠鈧�)), and the longer parallel side (also vertical) by the value of the function at the right subinterval endpoint (f(x岬�)). The height (horizontal length) of each trapezoid is the width of the subinterval (x岬� - x岬⑩倠鈧�).
04

Compute the Area of Each Trapezoid

To find the area of each trapezoid, use the formula: Area = ((shorter parallel side + longer parallel side) / 2) * height. For each trapezoid 饾憱 in the partition, its area is given by: ((f(x岬⑩倠鈧�) + f(x岬�)) / 2) * (x岬� - x岬⑩倠鈧�).
05

Sum the Areas of the Trapezoids

To approximate the definite integral, calculate the sum of the areas of all the trapezoids created in step 4: \begin{align*} \sum_{i=1}^{n}\frac{f(x_{i-1})+f(x_i)}{2}(x_i-x_{i-1}) \end{align*}.
06

Observe the Geometric Interpretation

Geometrically, the Trapezoid Rule can be seen as approximating the area under a curve by summing up the areas of trapezoids, which envelop the space between the curve and the x-axis. As the number of trapezoids (n) increases, the approximation becomes more accurate, converging to the value of the definite integral.

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