Problem 36 - Calculus

Question

If \(1 / \Delta x\) rectangles with base \(\Delta x\) touch the graph of \(u(x)\) at the left end of each interval (thus at \(x=0, \Delta x, 2 \Delta x, \ldots)\) express the total area in sigma notation.

Step-by-Step Solution

Answer

Use \( \sum_{n=0}^{\infty} u(n \Delta x) \Delta x \) to express the total area.

Step 1: Understanding the rectangle representation

The problem asks to find the total area of rectangles under the curve of a function. Each rectangle has a base of size \(\Delta x\) and the height is determined by the value of \(u(x)\) at the left endpoint of each interval which are \(x = 0, \Delta x, 2\Delta x, \ldots\).

Step 2: Define the nth interval

For each rectangle, \(n\) is an integer representing the interval number. So for the nth rectangle, the left endpoint is \(x = n \Delta x\).

Step 3: Express rectangle area formula

The area of the nth rectangle is the product of its base \(\Delta x\) and its height \(u(n \Delta x)\). Thus, the area can be written as \(u(n \Delta x) \Delta x\).

Step 4: Use sigma notation for total area

Since we are summing up the areas of infinitely many rectangles, we use sigma notation. The total area under the curve from \(x = 0\) can be expressed as: \[ \sum_{n=0}^{\infty} u(n \Delta x) \Delta x \] This sum represents the total area under the curve approximated by these rectangles.

Key Concepts

Sigma Notation

In mathematics, sigma notation is a concise way of representing the sum of a series. Sum symbols, or the Greek letter sigma (\(\Sigma\)), are used to show large quantities of numbers being added together. In this exercise, sigma notation is employed to denote the total area of rectangles under a curve. Each rectangle's area contributes to the overall sum of areas.
When using sigma notation, the formula \[ \sum_{n=0}^{\infty} a_n \] expresses the sum of terms \( a_n \) from \( n = 0 \) to infinity. In our case, \( a_n \) is the area \( u(n \Delta x) \Delta x\) of each rectangle.
By structuring the area as a function of \( n \), we efficiently sum the areas under the curve to approximate the total area with mathematical precision. This offers a systematic approach to summation in complex mathematical scenarios.

Rectangles Under Curve

The concept of approximating the area under a curve using rectangles is foundational in integral calculus. In this problem, each rectangle has a consistent base of \( \Delta x \) but varying height according to the function \( u(x) \). This height is determined at the left endpoint of each interval: points like \( 0, \Delta x, 2\Delta x \)..., which creates a series of vertical lines representing these left endpoints.
The height of a rectangle is the function value, \( u(n \Delta x) \), evaluated at these endpoints. Consequently, the higher the function value at a given point, the taller the corresponding rectangle becomes, influencing the total area calculation.

• Rectangles start at a specified point: the left endpoint.
• Their height is the function value at those points.
• Together, they approximate the area beneath the curve.

Area Approximation

Approximation of area under a curve using rectangles involves several steps and is a principle known as the Riemann Sum. By summing the area of each rectangle, we reach an approximate value for the total area under the graph of the function. This method helps in estimating integrals in calculus.
As the number of rectangles increases (by reducing \( \Delta x \)), the approximation becomes more accurate. Consequently, this series of approximations eventually approaches the definite integral of the function as \( \Delta x \to 0 \).

• More rectangles mean better accuracy.
• Reduction of \( \Delta x \) leads to a more precise approximation.
• The infinite sum of rectangles represents the integral of the function.

Function Intervals

Understanding the role of function intervals is pivotal when calculating areas using Riemann sums. Each interval is formed using a portion of the \( x \)-axis, divided into equal segments. In this exercise, these segments are determined by \( \Delta x \).
Each of these intervals contains a single rectangle, with the left endpoint defining the start of the interval. The position of the rectangle's base gives you where the function is evaluated to determine height.
Function intervals help in organizing the calculation by providing the structure necessary to apply consistent measurements for each rectangle.

• Intervals divide the \( x \)-axis into manageable segments.
• The left endpoint determines the rectangle’s height.
• These intervals contribute to a systematic area calculation.