Problem 22 - 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. $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(4-x_{k}^{* 2}\right) \Delta x_{k} \text { on }[-2,2]$$

Step-by-Step Solution

Answer

The given limit of Riemann sums can be expressed as a definite integral of the function \(f(x) = 4 - x^2\) on the interval \([-2, 2]\): $$\lim_{\Delta \rightarrow 0} \sum_{k=1}^{n}(4-x_{k}^{* 2})\Delta x_{k} = \int_{-2}^{2}(4 - x^2)\, dx$$

Step 1: Identify the Riemann Sum from the given limit

Given the limit, we can observe it is a limit of Riemann sums $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(4-x_{k}^{*2}\right) \Delta x_{k} \text { on }[-2,2]$$ Here, we can identify the function as \(f(x) = 4 - x^2\) based on the terms inside the summation, and the interval is \([a, b] = [-2, 2]\).

Step 2: Express the Riemann Sum as a Definite Integral

Now that we have identified the function \(f(x)\) and the interval, we can express the limit of Riemann sums as a definite integral. We know that $$\int_{a}^{b} f(x) \, dx = \lim _{\Delta \rightarrow 0} \sum_{k=1}^{n} f(x_{k}^*) \Delta x_{k}$$ Substituting the function and interval values, the definite integral becomes $$\int_{-2}^{2} (4-x^2) \, dx$$ So the given limit can be expressed as this definite integral: $$\lim _{\Delta \rightarrow 0} \sum_{k=1}^{n}\left(4-x_{k}^{* 2}\right) \Delta x_{k} = \int_{-2}^{2} (4 - x^2) \, dx$$

Key Concepts

Riemann Sum

When approaching the Riemann Sum, it is important to start from the basics. Riemann Sums are a way of approximating the area under a curve on a graph. Imagine you want to measure the area under a wiggly line. You would break it down into tiny rectangles that sit below the curve. The more rectangles you use, the better your approximation will be.
This method of summation is very useful:

• It divides the region under the curve into small, manageable pieces.
• Each piece is shaped like a tiny rectangle.
• Adding all these tiny rectangles together gives us an approximation of the area under the curve.

The Riemann Sum takes the form:\[ \sum_{k=1}^{n} f(x_{k}^*) \Delta x_{k} \]where:

• \(f(x_{k})\) represents the height of the function at each subinterval.
• \(\Delta x_k\) is the width of each subinterval.

As we reduce the width of these subintervals, the approximation becomes more accurate.

Limit of Sum

The Limit of Sum connects beautifully with the concept of Riemann Sums. In technical terms, it describes what happens when we let the number of rectangles reach infinity while their width approaches zero. Essentially, as the rectangles get smaller and more numerous, the Riemann Sum approaches the exact area under the curve.
We express this limit as: \[ \lim_{\Delta \rightarrow 0} \sum_{k=1}^{n} f(x_{k}^*) \Delta x_{k} \]Here’s what to keep in mind:

• It represents a transition from approximation to precise calculation.
• It is the mathematical basis for defining definite integrals.

So, by using Riemann Sums and taking their limit, we effectively calculate the exact area under a curve. This perfect harmony between approximation and limit showcases the beauty of calculus.

Function Identification

Function Identification is crucial when working with limits and integrals. In our exercise, the identification of the function from a Riemann Sum allows us to progress to solving the problem. Here’s how it works:
The given expression: \[ \sum_{k=1}^{n} \left(4-x_{k}^{*2}\right) \Delta x_{k} \]reveals that the function at play is \(f(x) = 4 - x^2\). Recognizing this function is vital because:

• It tells us how the function behaves over the interval.
• It helps in setting up the integral correctly.

Function identification allows us to bridge the gap between Riemann Sums and their corresponding definite integrals. Recognizing and writing down the function helps set the stage for integration over the defined range.

Integration

Integration is the grand finale of any journey involving Riemann Sums and Limits of Sums. Once the function and interval are identified, integration lets us calculate the exact area beneath a curve over a specific interval. With the identified function \(f(x) = 4 - x^2\) and the interval \([-2, 2]\), we can set up the integral as: \[ \int_{-2}^{2} (4 - x^2) \, dx \]Integration turns what once was an approximation into an exact value and provides answers in:

• A precise numerical form.
• Allows for the calculation of areas, volumes, and many other quantities.

Once the integral is set up, it can be solved using techniques such as antiderivatives. This process is fundamental in calculus and immensely powerful in various applications, from physics to engineering.