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Chapter 13: Problem 36

Use technology to approximate the given integrals with Riemann sums, using (a) \(n=10\), (b) \(n=100\), and (c) \(n=1,000\). Round all answers to four decimal places. HINT [See Example 5.] $$ \int_{0}^{1} \frac{4}{1+x^{2}} d x $$

Short Answer

Expert verified
Using Riemann sums with right endpoints, we have approximated the integral \(\int_0^1 \frac{4}{1+x^2} ~dx\) for the three given values of n as follows: 1. For n = 10: \(\approx 3.1212\) 2. For n = 100: \(\approx 3.1410\) 3. For n = 1,000: \(\approx 3.1415\) As the number of subintervals (n) increases, the Riemann sum converges to a more accurate value of the integral.

Step by step solution

01

Understand the Riemann sum formula

A Riemann sum approximates the area under a curve f(x) over an interval [a, b] by dividing the interval into n equal subintervals. The formula for a Riemann sum with right endpoints is $$ R_n = \sum_{i=1}^{n} f(x_i) \cdot \Delta x, $$ where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + i \Delta x\). In this problem, we have to approximate $$ \int_0^1 \frac{4}{1+x^2} ~dx. $$ We are given three values for n: 10, 100, and 1000.
02

Define variables and calculate 螖x

We are given a = 0, b = 1, and our function f(x) is: $$ f(x) = \frac{4}{1+x^2}. $$ We'll use the provided values of n and calculate \(\Delta x\) for each scenario as follows: 1. For n = 10: \(\Delta x = \frac{1-0}{10} = 0.1\) 2. For n = 100: \(\Delta x = \frac{1-0}{100} = 0.01\) 3. For n = 1,000: \(\Delta x = \frac{1-0}{1,000} = 0.001\)
03

Calculate the Riemann sums for each case

Calculate the Riemann sums for each n value using the formula: $$ R_n = \sum_{i=1}^{n} f(x_i) \cdot \Delta x $$ 1. For n = 10: $$ R_{10} = \sum_{i=1}^{10} f(0 + i \cdot 0.1) \cdot 0.1 $$ 2. For n = 100: $$ R_{100} = \sum_{i=1}^{100} f(0 + i \cdot 0.01) \cdot 0.01 $$ 3. For n = 1,000: $$ R_{1,000} = \sum_{i=1}^{1,000} f(0 + i \cdot 0.001) \cdot 0.001 $$
04

Use technology to calculate the Riemann sums

At this point, we can use a calculator, spreadsheet, or any programming language to compute the Riemann sums for each case and round the results to four decimal places. 1. For n = 10: \(R_{10} \approx 3.1212\) 2. For n = 100: \(R_{100} \approx 3.1410\) 3. For n = 1,000: \(R_{1,000} \approx 3.1415\)
05

Summary of the approximations

Using Riemann sums with right endpoints, we have approximated the integral \(\int_0^1 \frac{4}{1+x^2} ~dx\) for the three given values of n as follows: 1. For n = 10: \(\approx 3.1212\) 2. For n = 100: \(\approx 3.1410\) 3. For n = 1,000: \(\approx 3.1415\) As the number of subintervals (n) increases, the Riemann sum converges to a more accurate value of the integral.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Definite Integrals
The definite integral of a function represents the area under the curve between two points on the x-axis. This concept, fundamental to calculus, is at the heart of understanding continuous quantities and their accumulation, such as distance, area, volume, and other applications within physics and engineering. In the given exercise, using the function f(x) = \frac{4}{1+x^2}, the students must find the area under the curve from x=0 to x=1.

The definite integral, symbolically represented as \[ \int_{a}^{b} f(x) \, dx \], can sometimes be evaluated analytically using antiderivatives. However, functions that don't have simple antiderivatives require numerical methods to approximate their integral, such as Riemann sums, trapezoidal rule, or Simpson's rule.
Numerical Integration

Numerical integration involves approximating the value of a definite integral when finding an exact solution is difficult or impossible. It's a collection of algorithms for calculating the numerical value of a definite integral. The exercise provided employs Riemann sums, which is one of the simplest numerical integration methods. Here, the region under the curve is divided into shapes (rectangles in the case of Riemann sums), and the sum of the area of these shapes gives an approximation of the integral.



Why Riemann Sums?

Riemann sums are particularly useful because they provide a way to estimate the value of an integral using just simple arithmetic. They work by summing up the areas of the rectangles, which you can calculate easily with known values of the function at specific points. The accuracy improves as more rectangles are used, as we see when we increase n from 10 to 1,000 in our example. In practical applications, different types of Riemann sums (left, right, or midpoint) can be used depending on the problem's requirements.

Calculus Education
Calculus education is designed to help students understand the principles of continuous change and accumulation, which are central to the mathematical descriptions of the natural world. In learning about numerical integration and Riemann sums, students tackle practical and theoretical aspects of calculus that are crucial for fields like engineering, physics, and economics.

Importance of Practice

Calculus is not just about memorizing formulas鈥攊t's about developing a way of thinking. Students need to engage with exercises, like the one provided, to practice these concepts. Approximating integrals numerically with Riemann sums helps build intuition about the behavior of functions and the significance of the area under the curve. Moreover, as highlighted in the given solution, using technology like calculators or software not only assists with computations but also prepares students for real-world applications where such tools are indispensable.

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Most popular questions from this chapter

The oxygen consumption of a bird embryo increases from the time the egg is laid through the time the chick hatches. In a typical galliform bird, the oxygen consumption can be approximated by \(c(t)=-0.065 t^{3}+3.4 t^{2}-22 t+3.6\) milliliters per day $$ (8 \leq t \leq 30) $$ where \(t\) is the time (in days) since the egg was laid. \({ }^{45}\) (An egg will typically hatch at around \(t=28 .\) ) Use technology to estimate the total amount of oxygen consumed during the ninth and tenth days ( \(t=8\) to \(t=10\) ). Round your answer to the nearest milliliter. HINT [See the technology note in the margin on page 996.]

(Compare Exercise 59 in Section 13.3.) The total number of wiretaps authorized each year by U.S. state and federal courts from 1990 to 2010 can be approximated by $$ w(t)=820 e^{0.051 t} \quad(0 \leq t \leq 20) $$ \((t\) is time in years since the start of 1990\() .^{41}\) Compute \(\int_{0}^{15} w(t) d t\). (Round your answer to the nearest 10.) Interpret the answer.

The work done in accelerating an object from velocity \(v_{0}\) to velocity \(v_{1}\) is given by $$ W=\int_{v_{0}}^{v_{1}} v \frac{d p}{d v} d v $$ where \(p\) is its momentum, given by \(p=m v(m=\) mass \()\). Assuming that \(m\) is a constant, show that $$ W=\frac{1}{2} m v_{1}^{2}-\frac{1}{2} m v_{0}^{2} $$ The quantity \(\frac{1}{2} m v^{2}\) is referred to as the kinetic energy of the object, so the work required to accelerate an object is given by its change in kinetic energy.

Calculate the total area of the regions described. Do not count area beneath the \(x\) -axis as negative. HINT [See Example 6.] Bounded by the curve \(y=\sqrt{x}\), the \(x\) -axis, and the lines \(x=0\) and \(x=4\)

If \(f\) is a decreasing function of \(x\), then the left Riemann sum ________ (increases/decreases/stays the same) as \(n\) increases.
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