Problem 24 - Calculus Early Transcendentals

Question

Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function \(f\) on \([a, b] .\) Identify \(f\) and express the limit as a definite integral. $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left|x_{k}^{* 2}-1\right| \Delta x_{k} \text { on }[-2,2]$$

Step-by-Step Solution

Answer

Question: Express the given limit of the Riemann sum as a definite integral: $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left|x_{k}^{* 2}-1\right| \Delta x_{k}$$ for the function \(f(x) = |x^2 - 1|\) on the interval \([-2,2]\). Answer: $$\int_{-2}^2 |x^2 - 1|dx$$

Step 1: Identify the function and interval

In this exercise, our function, \(f(x)\), can be found inside the sum. The function is given as \(|x^2 - 1|\). The interval on which the function is defined is \([-2,2]\).

Step 2: Recognize the Riemann sum

A Riemann sum has the general form: $$\sum_{k=1}^{n} f(x^*_k)\Delta x_k$$ where \(f(x^*_k)\) is the function evaluated at some point \(x^*_k\) in the \(k\)-th subinterval, and \(\Delta x_k\) is the width of the \(k\)-th subinterval. The limit of the Riemann sum as the maximum width of the subintervals goes to 0 approximates the definite integral: $$\lim_{\Delta \rightarrow 0} \sum_{k=1}^{n} f(x^*_k) \Delta x_k = \int_a^b f(x) dx$$

Step 3: Express the limit as a definite integral

Now we can express the given limit of the Riemann sum as a definite integral: $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left|x_{k}^{* 2}-1\right| \Delta x_{k} = \int_{-2}^2 |x^2 - 1|dx$$

Step 4: Final Answer

The limit of the Riemann sums is equivalent to the definite integral: $$\int_{-2}^2 |x^2 - 1|dx$$

Key Concepts

Understanding Riemann Sums

A Riemann sum is a crucial concept in calculus that helps us approximate the area under a curve. To understand this, think of breaking the area under the curve of a function into small rectangles. Each rectangle's width is a small interval \(\Delta x_k\) along the x-axis. The height of each rectangle is determined by the value of the function at a specific point \(x^*_k\) within the interval.
The basic idea is simple:

• Subdivide the region into narrow strips or rectangles.
• Calculate the area of each rectangle using the formula for the area \(f(x^*_k) \times \Delta x_k\)
• The Riemann sum is then the sum of areas of all these rectangles.
• As the width of the rectangles — or \(\Delta\) — approaches zero, the Riemann sum becomes a closer approximation of the actual area.

By taking the limit of these sums as the width of the intervals approaches zero, we refine our approximation, eventually leading to what is known as a definite integral. This is why Riemann sums are so valuable—they give us a way to calculate integrals, providing a link between discrete sums and continuous calculus.

The Role of Function Evaluation in Calculus

Function evaluation is when we determine the value of a function at a particular point. In the context of Riemann sums and integral calculus, it is essential because it helps determine the height of the rectangles when approximating areas under a curve.
Whenever you see \(f(x^*_k)\), you're looking at function evaluation. Here:

• \(f(x^*_k)\) represents the function value at some representative point \(x^*_k\) within a subinterval.
• This tells you how high the rectangle should be at that particular section of the curve.
• Choosing different points within each subinterval can affect the accuracy of the Riemann sum.
• Common choices for \(x^*_k\) include the left endpoint, right endpoint, or midpoint of each subinterval.

Function evaluation is foundational in calculus because it allows us to analyze changes and behaviors of functions, guiding predictions and interpretations of real-world data.

Integral Calculus Explained

Integral calculus is all about accumulation. It focuses on finding functions that describe area, volume, and other concepts that accumulate over time or space. The definite integral is a core concept in integral calculus and is derived from the idea of the limit of Riemann sums.
Here's a breakdown:

• The process of integration is essentially adding up an infinite number of infinitesimally small quantities.
• Definite integrals \(\int_a^b f(x) \, dx\) give us the total sum of the function's accumulated values over an interval \([a, b]\).
• This calculation is closely related to the fundamental theorem of calculus, which links differentiation (finding rate of change) and integration (finding total accumulation).
• The definite integral computes not just area, but any scenario where a quantity accumulates, such as distance, mass, and probability.

Integral calculus provides the tools to handle real-world applications and solve complex problems, from physics to engineering, by accounting for continuous change and accumulation.