Question

Form the integral sum \(\mathrm{s}_{\mathrm{n}}\) by dividing the interval \([a, b]\) into \(n\) parts by the points \(x_{i}=a q^{i}(i=0,1,2,\), compute the following definite integrals: (i) \(\int_{a}^{b} x^{2} d x\) (ii) \(\int_{a}^{b} \frac{d x}{x}\), where \(0

Short answer

Question: Find the definite integrals for the following functions over the interval [a, b] with \(x_{i}=aq^{i}\) for \(i = 0,1,2,...\) and \(0<a<b\): (i) \(\int_{a}^{b} x^{2} dx\) (ii) \(\int_{a}^{b} \frac{dx}{x}\) (iii) \(\int_{a}^{b} \sqrt{x} dx\) Answer: (i) \(\int_{a}^{b} x^{2} dx = a^3(1-q)\lim_{n\to\infty}\sum_{i=1}^{n}q^{3i}\) (ii) \(\int_{a}^{b} \frac{dx}{x} = (1-q)\lim_{n\to\infty}\sum_{i=1}^{n} 1\) (iii) \(\int_{a}^{b} \sqrt{x} dx = a(1-q)\lim_{n\to\infty}\sum_{i=1}^{n}q^{i+\frac{1}{2}}\)

Step-by-step solution

Find the length of each sub-interval

Since the interval is divided into n equal parts, we need to find the length of each sub-interval, which is given by \(\Delta x_{i} = x_{i+1}-x_{i}=(aq^{i+1}-aq^{i})=aq^{i}(q-1)\). We'll use this in the next steps. Now, we'll find the definite integrals for each part.

Finding the integral sum for \(\int_{a}^{b}x^2dx\)

For (i), the integral sum Sn can be given as: \(s_{n} = \sum_{i=1}^{n}f(x_{i})\Delta x_{i} = \sum_{i=1}^{n} (aq^i)^2(1-q)aq^i\) \(s_{n} = a^3(1-q)\sum_{i=1}^{n}q^{3i}\) To find the definite integral, we'll take the limit as n approaches infinity: \(\int_{a}^{b} x^2 dx = \lim_{n\to\infty} s_{n} = a^3(1-q)\lim_{n\to\infty}\sum_{i=1}^{n}q^{3i}\) Now, we'll find the integral for (ii) using the same process.

Finding the integral sum for \(\int_{a}^{b}\frac{dx}{x}\)

For (ii), the integral sum Sn can be given as: \(s_{n} = \sum_{i=1}^{n}f(x_{i})\Delta x_{i} = \sum_{i=1}^{n} \frac{1}{aq^i}(1-q)aq^i\) \(s_{n} = (1-q)\sum_{i=1}^{n} 1\) To find the definite integral, we'll take the limit as n approaches infinity: \(\int_{a}^{b} \frac{dx}{x} = \lim_{n\to\infty} s_{n} = (1-q)\lim_{n\to\infty}\sum_{i=1}^{n} 1\) Finally, we'll find the integral for (iii).

Finding the integral sum for \(\int_{a}^{b}\sqrt{x}dx\)

For (iii), the integral sum Sn can be given as: \(s_{n} = \sum_{i=1}^{n}f(x_{i})\Delta x_{i} = \sum_{i=1}^{n} \sqrt{aq^i}(1-q)aq^i\) \(s_{n} = a(1-q)\sum_{i=1}^{n} q^{i+\frac{1}{2}}\) To find the definite integral, we'll take the limit as n approaches infinity: \(\int_{a}^{b} \sqrt{x} dx = \lim_{n\to\infty} s_{n} = a(1-q)\lim_{n\to\infty}\sum_{i=1}^{n}q^{i+\frac{1}{2}}\) So we have found the definite integrals for (i) \(\int_{a}^{b} x^{2} d x\), (ii) \(\int_{a}^{b} \frac{d x}{x}\) with the condition \(0<a<b\), and (iii) \(\int_{a}^{b} \sqrt{x} d x\).

Key concepts

Integral Calculus

Integral calculus is a fundamental component of calculus, focusing on the accumulation of quantities and the areas under curves. By definition, a definite integral represents the net area under a curve, within a boundary defined by an interval \[a, b\]. The process of finding this area requires the calculation of an antiderivative of the function and then evaluating it at the endpoints of the interval. However, when an antiderivative isn't readily obtainable, such as with non-elementary functions, numerical methods like integral sums are employed to approximate the definite integral.

For instance, consider the definite integral problem for the function \(x^2\) over the interval \[a, b\]. The solution involves dividing the interval into \(n\) equal parts, computing the sum of the areas of rectangles (integral sum), and taking the limit as the number of rectangles approaches infinity. This leads us to the precise area under \(x^2\) from \(a\) to \(b\). Understanding integral calculus not only aids in solving practical problems but also provides insights into the continuous accumulation of quantities such as distance, area, and volume.

Limit of a Sequence

The limit of a sequence is an essential concept in calculus, where you're often interested in the value a sequence of numbers approaches as the index (usually denoted as \(n\)) becomes very large. This concept underpins the calculation of many types of limits, including those used in finding definite integrals through Riemann sums.

In the context of integral calculus, when we work with sequences formed by integral sums, we take the limit of this sequence as \(n \to \infty\) to determine the definite integral. This is precisely what we do when calculating the area under a curve as a sequence of rectangles becomes infinitely narrow, essentially capturing more refined approximations of the area until it converges to the true value of the integral. For the given exercises, evaluating limits is essential for determining the exact value of the integrals for the functions \(x^2\), \(1/x\), and \(\sqrt{x}\).

Integral Sums

Integral sums, or Riemann sums, are used to approximate the area under a curve — a key element in calculating definite integrals. By splitting a curve into several subintervals and summing up the areas of rectangles or trapezoids that approximate the region under the curve, you form an integral sum. The height of each figure is usually determined by the value of the function at a specific point within the subinterval, known as the sample point.

In our given exercise, we calculate the integral sum for different functions. For each function, we take \(f(x_i)\) as the rectangle's height and \(\Delta x_i\) as its width. These sums vary depending on the function – whether it's \(x^2\), \(1/x\), or \(\sqrt{x}\). The integral sums provide a numerical approximation, which becomes more accurate as the number of subintervals increases and their width decreases. Therefore, to achieve the definite integral, we consider the limit of these sums as the number of intervals \(n\) approaches infinity.

Riemann Sum

A Riemann sum is a specific example of an integral sum and one of the primary concepts for understanding definite integrals. Named after the German mathematician Bernhard Riemann, a Riemann sum estimates the area under a curve by summing up areas of shapes that approximate the region; these shapes can be rectangles, trapezoids, or others. The choice of points where the function is evaluated within each subinterval leads to different types of Riemann sums: left, right, midpoint, or trapezoidal, among others.

The exercises provided illustrate Riemann sums for several simple power functions. By calculating the area of rectangles at each point \(x_i\), and by taking the sum of all these areas, we get a Riemann sum approximation of the integral. As the number of rectangles increases, the width of each rectangle becomes smaller, and the approximation improves. The true area— the definite integral — is the limit of these Riemann sums as the number of rectangles approaches infinity and their width approaches zero.