Problem 4 - Thomas Calculus

Question

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\frac{1}{c_{k}}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[1,4] $$

Step-by-Step Solution

Answer

The limit is expressed as \(\int_{1}^{4} \frac{1}{x} \, dx\).

Step 1: Identify the interval and function

The given limit refers to a Riemann sum on the interval \([1, 4]\). Each term in the sum involves \(\frac{1}{c_k}\), where \(c_k\) is a sample point in the subinterval. The function of interest here is \(f(x) = \frac{1}{x}\).

Step 2: Recognize the Riemann Sum

The expression \(\sum_{k=1}^{n}\left(\frac{1}{c_{k}}\right) \Delta x_{k}\) represents a Riemann sum for the function \(f(x) = \frac{1}{x}\) over the interval \([1, 4]\). As \(||P|| \rightarrow 0\), this sum converges to the definite integral of \(f\).

Step 3: Convert the Riemann Sum to a Definite Integral

The limit of the Riemann sum as the norm of the partition approaches zero becomes the definite integral of the function over the given interval. Thus, we have: \[\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}\left(\frac{1}{c_{k}}\right) \Delta x_{k} = \int_{1}^{4} \frac{1}{x} \, dx\]

Step 4: Write the final expression

Hence, the limit expression given in the problem can be rewritten as the definite integral: \[\int_{1}^{4} \frac{1}{x} \, dx\]

Key Concepts

Riemann Sum

In calculus, a Riemann sum is a crucial concept used to approximate the area under a curve. This area is between the curve itself and the x-axis over an interval. Riemann sums allow us to approximate integrals, rooted in dividing the interval of interest into smaller subintervals. Each subinterval is associated with a sample point, denoted as \(c_k\). The function value at each \(c_k\) is then multiplied by the width of its corresponding subinterval, \(\Delta x_k\). Consequently, adding all these small areas gives what is known as the Riemann sum:

• Approximation: Riemann sums provide a method to approximate the integral by summing up areas of rectangles covering the curve.
• Flexibility: The choice of sample points (\(c_k\)) within each subinterval can vary, offering different types of Riemann sums like left, right, and midpoint sums.

The finer the partition, which means the smaller the \(\Delta x_k\), the closer the sum becomes to the actual integral. This convergence is captured by taking the limit of the Riemann sum as described when \(\|P\| \rightarrow 0\).

Function Analysis

Function analysis is the process of examining a function to understand its behavior over a certain interval. Essentially, function analysis allows us to predict how a function will act, which facilitates the identification of specific function values on particular intervals. In this case, the function given is \(f(x) = \frac{1}{x}\). It's crucial to understand how this function behaves:

• Domain: For \(f(x) = \frac{1}{x}\), the domain is all real numbers except \(x = 0\) since division by zero is undefined.
• Behavior over [1, 4]: Here, the function is decreasing but always positive, meaning it approaches closer to zero as \(x\) increases, but never reaches it.

This analysis helps in both setting the correct bounds for our integral and understanding how changes in \(x\) affect the function's value. A thorough function analysis is pivotal before converting a Riemann sum into a definite integral.

Limit of Partition

The concept of the limit of partition revolves around creating smaller and smaller subdivisions of an interval to find the exact value of an integral. In simpler terms, it involves refining the partition or the breaking down of the interval over which a function is integrated. When we say \(\|P\| \rightarrow 0\), we mean that the maximum width of all subintervals in the partition is becoming infinitesimally small:

• Precision: As \(\|P\|\) approaches zero, the Riemann sum becomes more accurate, converging to the actual value of the definite integral.
• Continuity of changes: The transitions in the size of the partitions are continuous, leading to better approximation.

This concept is central to the notion of integration itself, which seeks to find the limit of these sums through infinitely refining the partitions. It ensures that Riemann sums resolve to definite integrals, the ultimate objective of which is to find the exact area under a curve over a specified interval.