Problem 3 - Thomas Calculus

Question

Express the limits in Exercises \(1-8\) as definite integrals. $$ \lim _{|P| \rightarrow 0} \sum_{k=1}^{n}\left(c_{k}^{2}-3 c_{k}\right) \Delta x_{k}, \text { where } P \text { is a partition of }[-7,5] $$

Step-by-Step Solution

Answer

\(\int_{-7}^{5} (x^{2} - 3x) \; dx\)

Step 1: Identify the function inside the sum

The expression inside the sum is \(c_{k}^{2}-3c_{k}\). This represents the function we need to integrate.

Step 2: Understand the definition of Riemann Sums

Recall that the limit of a Riemann sum as the partition size \(|P|\) approaches zero represents the definite integral of the function over the interval. In this case, \(|P|\) is the norm of the partition of the interval \([-7, 5]\).

Step 3: Write the limit as a definite integral

Since \(c_k\) represents a sample point in each subinterval of the partition of \([-7, 5]\), the limit of the sum \(\sum_{k=1}^{n}(c_{k}^{2} - 3c_{k})\Delta x_{k}\) as \(|P| \rightarrow 0\) can be expressed as the definite integral:\[\int_{-7}^{5} (x^{2} - 3x) \; dx\]

Key Concepts

Riemann Sum

A Riemann sum is a method used to approximate the area under a curve, which can be translated into evaluating definite integrals. When breaking down the concept of a Riemann sum, think of it as carving a continuous space (under a curve) into small slices, or subintervals. Each of these subintervals has a width, typically labelled as \(\Delta x_k\), and each has an associated height, given by the value of the function at some point within that subinterval.

• The Riemann sum is the total of the areas of these rectangles that approximate the area under the curve.
• Mathematically, it looks like \(\sum_{k=1}^{n} f(c_k) \Delta x_k\), where \(f(c_k)\) is the function value at the chosen point \(c_k\) of each subinterval.

As the width goes to zero, or the partitions become finer, the Riemann sum approaches the exact area under the curve, thus leading to a definite integral.

Partition Interval

The partition interval is crucial in the composition of a Riemann sum. It refers to the process of breaking an interval into smaller subintervals, which are used to evaluate how the function behaves over the entire range.

• Consider an interval, in this case, \([-7, 5]\). Here, we partition this interval into \(n\) subintervals.
• The subintervals may not always be equal in size; however, as the Riemann sum becomes a definite integral, the subintervals tend to be smaller and smaller in a limiting process.

By making each subinterval smaller, we can better approximate the function's behavior over the interval, making the partition finer improves the approximation's accuracy. This is represented mathematically as \(|P| \to 0\). The norm \(|P|\) of the partition is the length of the longest subinterval in the partition.

Definite Integral Representation

Definite integral representation is the outcome of taking the limit of a Riemann sum whose partitions become infinitely fine. This process converts the sum of small areas into an integral notation, giving an exact area under a curve.

• In a definite integral, we write \(\int_{a}^{b} f(x) \, dx\), where \(\int\) symbolizes the integral, the limits of integration \(a\) and \(b\) represent the bounds of the area under consideration, and \(f(x)\) is the function to be integrated.
• For our problem, the integral \(\int_{-7}^{5} (x^{2} - 3x) \, dx\) depicts the exact area under the curve from \(x = -7\) to \(x = 5\).

Through definite integrals, we can accurately calculate areas, volumes, and other important quantities in calculus, making them essential in both theoretical and applied mathematics.