Problem 8 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The limit is expressed as the integral \(\int_{0}^{\pi/4} \tan(x)\, dx\).

Step 1: Understand the Limit Representation

The given expression is the limit of a Riemann sum. It's expressed as \( \lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n}(\tan c_{k}) \Delta x_{k} \), where \( P \) is a partition of the interval \([0, \pi/4]\). This represents the integral of the function \( f(x) = \tan(x) \) over that interval.

Step 2: Identify the Definite Integral

Recognize that in the context of Riemann sums, as \( \|P\| \rightarrow 0 \) (meaning the maximum width of the partition intervals approaches zero), the sum \( \sum_{k=1}^{n}(\tan c_{k}) \Delta x_{k} \) becomes the integral of \( f(x) \) from the lower to the upper limit of the partition. Thus, the limit corresponds to the definite integral \( \int_{0}^{\pi/4} \tan(x)\, dx \).

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the total area underneath a curve, or more generally, the integral of a function over an interval. The idea is to break up the interval into smaller sections, or subintervals, then calculate the area for each section using simple geometric shapes, such as rectangles.
The sum of these rectangle areas gives an approximation of the integral.

• The interval \([a, b]\) is divided into \([x_0, x_1, x_2, ... , x_n]\).
• For each subinterval \([x_{k-1}, x_k]\), let \(c_k\) be a point in this subinterval.
• The Riemann sum is then \(\sum_{k=1}^{n}f(c_k)\Delta x_k\), where \(\Delta x_k = x_k - x_{k-1}\).

In the context of the given exercise, each \(c_k\) represents a point within each subinterval of the partition \(P\), and \(f(c_k) = \tan(c_k)\) where the function being considered is \(\tan(x)\). As the partitions become finer, this sum becomes a more accurate approximation of the total integral.

Limit of a Partition

The concept of the 'limit of a partition' deals with refining our approximation methods to an exact mathematical result. To find an exact integral, we make the subintervals as narrow as possible.
This means the maximum width of the subintervals, \(\|P\|\), approaches zero. This process is described as taking the limit of the norm of the partition to 0.

• As \(\|P\| \to 0\), we can find the precise value of the area under the curve.
• The Riemann sum turns into a definite integral when the partitions are infinitely fine.

In the given exercise, this limiting process transforms the Riemann sum into the definite integral \(\int_{0}^{\pi/4} \tan(x)\, dx\). This tells us the exact area under \(\tan(x)\) from \(0\) to \(\pi/4\).

Integration

Integration is one of the two main operations of calculus, the other being differentiation. It is used to find the accumulation of quantities, such as areas under curves, volumes, and other sums.
In a definite integral, the focus is on calculating accumulation over a specific interval, \([a, b]\).

• A definite integral is denoted as \(\int_{a}^{b}f(x)\,dx\), where \(f(x)\) is the function being integrated.
• This operation results in a single numerical value that represents the total accumulation between the starting and ending points.

In our problem, the integral \(\int_{0}^{\pi/4} \tan(x)\, dx\) is a definite integral. This involves integrating the function \(\tan(x)\) over the interval \(\left[0, \frac{\pi}{4}\right]\). Here, \(\tan(x)\) is a trigonometric function, adding complexity to the integration process but providing clear insight into the behavior of this function over the specified interval.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, describing the relationships between the angles and sides of triangles, and extending these definitions to the unit circle.
These functions also turn up frequently in various branches of science and engineering.

• Common trigonometric functions include \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\).
• They are periodic and have specific unique properties that can complicate calculations but offer rich applications and insights.

In the context of the given exercise, \(\tan(x)\) — the tangent function — is integrated over the interval \([0, \pi/4]\). \(\tan(x)\) is defined as the ratio of \(\sin(x)\) to \(\cos(x)\), and it has a period of \(\pi\). The integration of \(\tan(x)\) involves understanding these properties for correct evaluation and application in the definite integral.