Problem 6 - Thomas Calculus

Question

Express the limits in Exercises \(1-8\) as definite integrals. \begin{equation} \lim _{ \| P | \rightarrow 0} \sum_{k=1}^{n} \sqrt{4-c_{k}^{2}} \Delta x_{k}, \text { where } P \text { is a partition of }[0,1] \end{equation}

Step-by-Step Solution

Answer

The limit is expressed as the integral \(\int_{0}^{1} \sqrt{4-x^2} \, dx.\)

Step 1: Identify the Integral Components

Recognize that the given limit is a Riemann sum, a representation of the definite integral. The expression \( \sqrt{4-c_k^2} \Delta x_k \) is the Riemann sum for a function over an interval \([0,1]\). We need to express this as a definite integral.

Step 2: Recognize the Function and Interval

The function from the Riemann sum is \(f(x) = \sqrt{4-x^2}\) evaluated at \(c_k\). The interval is clearly \([0, 1]\) since that is the partition specified for \(P\).

Step 3: Setup the Definite Integral

The Riemann sum \(\sum_{k=1}^{n} \sqrt{4-c_{k}^{2}} \Delta x_{k}\) becomes a definite integral as the partition becomes finer (i.e., as \(||P|| \to 0\)). Thus, the limit can be expressed as the integral of the function over the specified interval: \[\int_{0}^{1} \sqrt{4-x^2} \, dx.\]

Step 4: Finalize the Integral Expression

The final expression for the limit in terms of a definite integral over the interval [0,1] is \[\int_{0}^{1} \sqrt{4-x^2} \, dx.\] This is the continuous form representation of the Riemann sum.

Key Concepts

Riemann Sum

A Riemann sum is a method for approximating the total value of a function over an interval. It does this by breaking the interval into smaller sub-intervals or partitions. For each sub-interval, you take a sample point, calculate the function's value there, and multiply it by the width of the sub-interval. This value approximates the area under the curve for that segment.

By adding the areas of all these sub-intervals together, you get the Riemann sum, which approaches the integral as the width of sub-intervals approaches zero.

In the exercise, the expression \( \sqrt{4-c_k^2} \, \Delta x_k \) represents the Riemann sum for the function \( f(x) = \sqrt{4-x^2} \) over the interval \([0,1]\). As \(|P| \rightarrow 0\), this sum becomes a more accurate approximation of the definite integral of the function over the interval.

Interval Notation

Intervals are used to describe a set of real numbers lying between two endpoints. Interval notation is a handy tool for expressing such sets concisely and clearly.

The notation has two forms:

• Closed interval \([a, b]\), where both \(a\) and \(b\) are included in the set.
• Open interval \((a, b)\), where \(a\) and \(b\) are not included.

The given Riemann sum in the exercise is defined over the closed interval \([0, 1]\). This tells us that the integration includes both the starting number, 0, and the ending number, 1.

Function Representation

Function representation is the process of expressing a mathematical function with an equation, graph, or other means. In calculus, functions describe relationships between two variables, typically \(x\) and \(y\).

The function presented in the exercise is \(f(x) = \sqrt{4-x^2}\). This is a semicircle with a radius of 2 and is part of the equation of a circle \(x^2 + y^2 = 4\). This particular function only describes the upper half of the circle. By examining this function over the interval \([0,1]\), we look at part of this semicircle and calculate the area beneath it for that portion.

Integral Calculus

Integral calculus is a branch of calculus focused on finding the total accumulation of a quantity. For functions, this often involves calculating the area under a curve.

The definite integral, denoted by \(\int_{a}^{b} f(x) \, dx\), represents the accumulation of the function \(f(x)\) from \(x = a\) to \(x = b\). Definite integrals combine limits and summation to calculate precise quantities.

In the exercise, the Riemann sum is transformed into a definite integral, \(\int_{0}^{1} \sqrt{4-x^2} \, dx\). This represents the exact area under the curve of \(f(x) = \sqrt{4-x^2}\) from \(x = 0\) to \(x = 1\). It is a crucial technique in calculus for solving real-world problems, where precise calculations are necessary.