Problem 35 - Calculus

Question

Suppose \(n\) rectangles with base \(\Delta x\) touch the graph of \(u(x)\) at the points \(x=\Delta x, 2 \Delta x, \ldots, n \Delta x\). Express the total rectangular area in sigma notation.

Step-by-Step Solution

Answer

Total area is \( \sum_{i=1}^{n} u(i \Delta x) \cdot \Delta x \).

Step 1: Understanding the Problem

You are asked to express the area of rectangles under the curve described by the function \( u(x) \), where the rectangles touch the curve at certain points: \( x = \Delta x, 2\Delta x, \ldots, n \Delta x \). The width of each rectangle is given as \( \Delta x \).

Step 2: Identify Rectangular Area Formula

Each rectangle has a width \( \Delta x \) and a height equal to the value of \( u(x) \) at specific points. Thus, the area of each rectangle can be expressed as \( A_i = u(i \Delta x) \cdot \Delta x \) for \( i = 1, 2, \ldots, n \).

Step 3: Summing the Areas

The total area is the sum of the areas of all rectangles. This can be written as \( A_{total} = \sum_{i=1}^{n} u(i \Delta x) \cdot \Delta x \), where the summation symbol \( \Sigma \) is used to denote the sum of each rectangular area from \( i = 1 \) to \( n \).

Step 4: Express in Sigma Notation

The total area under the curve, as approximated by these rectangles, is expressed in sigma notation as follows:\[A_{total} = \sum_{i=1}^{n} u(i \Delta x) \cdot \Delta x\]

Key Concepts

Sigma Notation

Sigma notation is a compact way to represent the sum of many similar terms. Instead of writing long addition sequences, we use the Greek letter Sigma (\(\Sigma\)) to indicate that the terms should be summed up. The general form looks like this:
\[\sum_{i=a}^{b} f(i)\]where \\(i\) is called the index of summation, \\(a\) is the lower limit of summation, and \\(b\) is the upper limit. Each value of \\(i\) within the limits is substituted into the function \\(f(i)\), and all resulting values are added together.

• The notation is helpful when dealing with large sums that follow a predictable pattern.
• It provides a clear way to represent "adding up" over a range.

In the context of the exercise, sigma notation is used to sum up the areas of rectangles touching a curve. Each term \\(u(i \Delta x) \cdot \Delta x\) represents an individual rectangle's area, and the entire expression neatly captures all such terms by using sigma notation.

Rectangular Approximation

Rectangular approximation is an essential method when estimating the area under a curve. This method uses rectangles to approximate the area between a curve and the x-axis. Suppose you have a function curve \\(u(x)\), and you want to estimate the area under it between certain points.

• Choose a number of rectangles (say, \\(n\)), which will fit beneath the curve.
• Determine the width of each rectangle (\(\Delta x\)).

The height of each rectangle in this approximation is determined by the value of the function at specific x-values, such as \\(x = i \Delta x\)n (i.e., points like \\(\Delta x, 2\Delta x, \ldots\, n\Delta x\)). The area of each rectangle is therefore \\(u(i \Delta x) \cdot \Delta x\)n.
Benefits of this method include that it provides a simple way to understand integral calculus practically, as the sum of these areas becomes a Riemann sum, a foundation for understanding definite integrals.

Definite Integrals

Definite integrals are closely linked to the concept of finding the area under curves. When you have a continuous function, the definite integral from \\(a\) to \\(b\) represents the net area between the function and the x-axis over that interval.

• The notation for a definite integral is \\(\int_{a}^{b} f(x)\, dx\).
• It provides a precise value for the area under a curve or between curves.

This contrasts with a Riemann sum, which gives an approximation by adding up areas of rectangles under the curve. As the number of rectangles (\(n\)) increases in a Riemann sum \\((\sum_{i=1}^{n} u(i \Delta x) \cdot \Delta x)\), and \\(\Delta x\) tends to zero, the approximation improves, eventually converging to the exact value represented by the definite integral. Thus:\[\lim_{n \to \infty} \sum_{i=1}^{n} u(i \Delta x) \cdot \Delta x = \int_{a}^{b} u(x)\, dx\]Definite integrals are a critical component in calculus for precisely calculating areas, and they help in understanding the behavior of functions over intervals.