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Chapter 7: Problem 2

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

Short Answer

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Question: Explain geometrically how the Midpoint Rule is used to approximate a definite integral. Answer: The Midpoint Rule approximates a definite integral geometrically by subdividing the interval into equal subintervals and approximating the area under the curve using rectangles. The height of each rectangle is determined by the value of the function at the midpoint of each subinterval. By summing the area of all rectangles, we approximate the total area under the curve over the given interval. The larger the value of the number of subintervals (n), the better the approximation.

Step by step solution

01

Defining the Midpoint Rule

The Midpoint Rule is a numerical integration technique used to approximate the definite integral of a function over a given interval. The method works by subdividing the interval into equal subintervals and approximating the area under the curve over each subinterval using rectangles. The height of each rectangle is determined by the value of the function at the midpoint of the subinterval.
02

Geometric interpretation of a definite integral

A definite integral represents the area under a curve (f(x)) over the interval [a, b]. The area can be thought of as the accumulation of infinitely many infinitesimally small quantities dx, each representing the area of a thin strip between the curve and the x-axis. Geometrically, approximating a definite integral is equivalent to approximating the area under the curve.
03

Dividing the interval

To start the Midpoint Rule, we first divide the interval [a, b] into n equal subintervals, each of width 螖x = (b - a) / n. Suppose x_i is the beginning point of each subinterval and x_(i+1) is the end point, and x* is the midpoint of the subinterval. Then x* = (x_i + x_(i+1)) / 2.
04

Approximating the area using rectangles

For each subinterval, we approximate the area under the curve using a rectangle. The height of each rectangle is determined by the value of the function at the midpoint of that subinterval (螖x * f(x*)). By summing the area of all rectangles, we approximate the total area under the curve over the interval [a, b].
05

Midpoint Rule formula

The Midpoint Rule can be expressed mathematically as follows: integral(a to b) f(x) dx 鈮� 螖x * (f(x_1*) + f(x_2*) + ... + f(x_n*)) The larger the value of n (the number of subintervals), the better the approximation.
06

Geometrical explanation

In conclusion, the Midpoint Rule approximates the definite integral geometrically by subdividing the interval into equal subintervals and approximating the area under the curve using rectangles, with the height determined by the value of the function at the midpoint of each subinterval. This method provides a reasonable approximation for the area under a curve and forms the basis for more sophisticated numerical integration techniques.

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