Problem 22 - A History of Mathematics: An Introduction

Question

Determine the area under the curve \(y=p x^{k}\) from \(x=0\) to \(x=x_{0}\) by dividing the interval \(\left[0, x_{0}\right]\) into an infinite set, of subintervals, beginning from the right with the points \(a_{0}=x_{0}, a_{1}=\frac{n}{m} x_{0}, a_{2}=\left(\frac{n}{m}\right)^{2} x_{0}, \ldots\), where \(n

Step-by-Step Solution

Answer

Question: Determine the area under the curve \(y = px^k\) from \(x = 0\) to \(x = x_0\) using the given geometric progression of subintervals. Answer: The area under the curve \(y = px^k\) from \(x = 0\) to \(x = x_0\) is given by \(A(x_0) = x_0p\left(\frac{\left(\frac{n}{m}\right)}{1-\left(\frac{n}{m}\right)}\right)^{k+1}\).

Step 1: Geometric interpretation of the problem

In order to find the area under the curve, we will approximate this area with a sum of areas of rectangles. The width of these rectangles starts with \(a_0=x_0\) and decreases by the geometric progression with a common ratio \(\frac{n}{m}<1\), with \(n

Step 2: General expression for the area of rectangles

The width of the rectangles is given by the difference \(a_i - a_{i-1}\), where \(a_i = \left(\frac{n}{m}\right)^i x_0\). For each rectangle, the height is given by the value of the function \(y = px^k\) at the point \(x=a_i\). We can write this as \(y(p, a_i)=pa_i^k\). Thus, the area of each rectangle can be written as: $$ A_i = (a_{i-1} - a_i) y(p, a_i) = \left(\left(\frac{n}{m}\right)^{i-1}x_0 - \left(\frac{n}{m}\right)^i x_0\right)(p\left(\frac{n}{m}\right)^{ik}) $$

Step 3: Evaluate the limit of the sum of the areas

To find the area under the curve, we need to sum the areas of these rectangles and take the limit as the number of rectangles goes to infinity: $$ A(x_0) = \lim_{N\to\infty} \sum_{i=1}^N A_i = \lim_{N\to\infty} \sum_{i=1}^N \left[\left(\frac{n}{m}\right)^{i-1}x_0 - \left(\frac{n}{m}\right)^i x_0\right](p\left(\frac{n}{m}\right)^{ik}). $$ Since this is a geometric series, we can use the formula for the sum of an infinite geometric series: $$ \sum_{i=1}^{\infty} \left(\frac{n}{m}\right)^i = \frac{\left(\frac{n}{m}\right)}{1-\left(\frac{n}{m}\right)}. $$ Thus, the area under the curve is: $$ A(x_0) = x_0p\sum_{i=1}^{\infty} \left[\left(\frac{n}{m}\right)^{i-1} - \left(\frac{n}{m}\right)^i\right] \left(\frac{n}{m}\right)^{ik} = x_0p\left(\frac{\left(\frac{n}{m}\right)}{1-\left(\frac{n}{m}\right)}\right)^{k+1}. $$ Therefore, the area under the curve \(y = px^k\) from \(x=0\) to \(x=x_0\) is given by: $$ A(x_0) = x_0p\left(\frac{\left(\frac{n}{m}\right)}{1-\left(\frac{n}{m}\right)}\right)^{k+1}. $$

Key Concepts

Understanding the Area Under the Curve

When it comes to calculus, one of the essential concepts is the 'area under the curve'. This represents the region bounded by the graph of a function, the x-axis, and the vertical lines corresponding to the limits of integration. In practical terms, finding the area under the curve can signify anything from the total distance traveled by an object in motion (when the curve represents its speed over time), to the amount of resources consumed over a period.

To determine this area with precision, one typically employs integral calculus. However, for curves that lack simple antiderivatives or when dealing only with basic tools, we approximate by summing the areas of shapes, such as rectangles or trapezoids, that fit under the curve. This method is known as Riemann sums, and by refining the number of these shapes to infinity, we can obtain an increasingly accurate approximation of the true area.

In our exercise, the area under the curve is approached by creating a series of rectangles whose combined area approaches the true area as the number of rectangles approaches infinity. Since the widths of these rectangles form a geometric progression, the problem nicely segues into exploring geometric series and their convergence.

Grasping the Concept of Infinite Series

Infinite series are foundational in understanding how to sum an endless list of numbers. Imagine adding up an infinite number of terms; it seems impossible at first glance. However, through the concept of convergence, certain infinite series can sum to a finite value. This is critical in mathematics because it underpins various analysis, calculus, and applied mathematics fields.

The infinite series can be shown as a sum of terms that, while inexhaustible, approach a limit. This brings us to the series in our geometric progression problem. When the common ratio is between -1 and 1 (excluding the endpoints), the infinite series will converge to a precise value, allowing us to compute the aggregate of an infinite number of terms - a seemingly paradoxical but fascinating and useful mathematical reality.

In our textbook example, we apply the concept of infinite series to find the exact sum of the areas of an infinite number of rectangles under a curve, demonstrating a practical application of abstract mathematical theory.

Diving Into Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric progressions are powerful tools in both algebra and calculus for modeling exponential growth or decay, such as population growth, radioactive decay, and interest calculations.

In the context of the provided exercise, the width of each successive rectangle under the curve shrinks according to a geometric progression, with a common ratio that is less than 1. This is because each term is a fraction of the previous term, representing a decrease. The importance of understanding geometric progression lies in its relation to the solution of the exercise: the sum of the areas of rectangles, which are determined by the terms of the sequence, yields the total area under the curve as the number of terms grows infinitely large.

Understanding geometric progressions not only assists in solving problems involving series, but also enriches our comprehension of how repeated multiplication can lead to powerful expressions of natural phenomena and financial scenarios.

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Problem 21

Next

Problem 23

Other exercises in this chapter

Problem 18

Show that Fermat's rule, $$ N\left(\begin{array}{c} N+k \\ k \end{array}\right)=(k+1)\left(\begin{array}{c} N+k \\ k+1 \end{array}\right) $$ is equivalent to $$

View solution

Problem 21

Fermat included the following result in a letter to Roberval dated August 23, 1636: If the parabola with vertex \(A\) and axis \(A D\) is rotated around the lin

View solution

Problem 23

Using Wallis's method, interpolate the row \(p=3\) in his ratio table for \(n=1 / 2, n=3 / 2\), and \(n=5 / 2\).

View solution

Problem 25

Find the length of arc of the curve \(y^{4}=x^{5}\) from \(x=0\) to \(x=b\).

View solution

Key Concepts

Understanding the Area Under the Curve

When it comes to calculus, one of the essential concepts is the 'area under the curve'. This represents the region bounded by the graph of a function, the x-axis, and the vertical lines corresponding to the limits of integration. In practical terms, finding the area under the curve can signify anything from the total distance traveled by an object in motion (when the curve represents its speed over time), to the amount of resources consumed over a period.

To determine this area with precision, one typically employs integral calculus. However, for curves that lack simple antiderivatives or when dealing only with basic tools, we approximate by summing the areas of shapes, such as rectangles or trapezoids, that fit under the curve. This method is known as Riemann sums, and by refining the number of these shapes to infinity, we can obtain an increasingly accurate approximation of the true area.

In our exercise, the area under the curve is approached by creating a series of rectangles whose combined area approaches the true area as the number of rectangles approaches infinity. Since the widths of these rectangles form a geometric progression, the problem nicely segues into exploring geometric series and their convergence.

Grasping the Concept of Infinite Series

Infinite series are foundational in understanding how to sum an endless list of numbers. Imagine adding up an infinite number of terms; it seems impossible at first glance. However, through the concept of convergence, certain infinite series can sum to a finite value. This is critical in mathematics because it underpins various analysis, calculus, and applied mathematics fields.

The infinite series can be shown as a sum of terms that, while inexhaustible, approach a limit. This brings us to the series in our geometric progression problem. When the common ratio is between -1 and 1 (excluding the endpoints), the infinite series will converge to a precise value, allowing us to compute the aggregate of an infinite number of terms - a seemingly paradoxical but fascinating and useful mathematical reality.

In our textbook example, we apply the concept of infinite series to find the exact sum of the areas of an infinite number of rectangles under a curve, demonstrating a practical application of abstract mathematical theory.

Diving Into Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Geometric progressions are powerful tools in both algebra and calculus for modeling exponential growth or decay, such as population growth, radioactive decay, and interest calculations.

In the context of the provided exercise, the width of each successive rectangle under the curve shrinks according to a geometric progression, with a common ratio that is less than 1. This is because each term is a fraction of the previous term, representing a decrease. The importance of understanding geometric progression lies in its relation to the solution of the exercise: the sum of the areas of rectangles, which are determined by the terms of the sequence, yields the total area under the curve as the number of terms grows infinitely large.

Understanding geometric progressions not only assists in solving problems involving series, but also enriches our comprehension of how repeated multiplication can lead to powerful expressions of natural phenomena and financial scenarios.