Problem 2 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

In summary, the Midpoint Rule is a geometric technique used to approximate the area under a curve (represented by a definite integral) by using the midpoints of sub-intervals created by partitioning the given interval. It computes the area by summing up the areas of rectangles with heights determined by the function's value at the midpoint of each sub-interval and with widths equal to the width of each partition. The accuracy of the approximation improves as the number of partitions (n) increases.

Step 1: Understanding the definite integral

A definite integral represents the area under a curve for a specific interval. Mathematically, it is defined as: \[ \int_a^b f(x) dx \] where f(x) is the function being integrated, and 'a' and 'b' are the lower and upper limits of the interval.

Step 2: Partitioning the interval of integration

To approximate the area under the curve, we can partition the interval `[a, b]` into 'n' equal sub-intervals. Each of these sub-intervals will have a width 'Δx' where: \[ Δx = \frac{b - a}{n} \] These partitions will help create rectangles, whose area will sum up to approximate the area under the curve.

Step 3: Defining the Midpoint Rule and its geometric interpretation

The Midpoint Rule approximates the definite integral by using the midpoints of each partition to compute the area of the rectangles. The heights of the rectangles are determined by the value of the function at the midpoint of the sub-interval, represented as: \[ M_n = Δx \left[ f \left( \frac{a + a + Δx}{2} \right) + f \left( \frac{a + Δx + a + 2Δx}{2} \right) + \cdots + f \left( \frac{a + (n-1)Δx + a + nΔx}{2} \right)\right] \] Geometrically, the Midpoint Rule uses the midpoints of the sub-intervals to create rectangles whose total area is an approximation of the area under the curve. The accuracy of the approximation improves as the number of partitions 'n' increases.