/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Why might an integral found in a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 3

Why might an integral found in a table differ from the same integral evaluated by a computer algebra system?

Short Answer

Expert verified
Answer: The possible reasons for the difference between an integral value found in a table and the same integral evaluated by a computer algebra system are: 1. Approximations in the table 2. Different integration techniques 3. Different forms of the integral 4. Errors in the table 5. Numerical instabilities in the computer system

Step by step solution

01

Possible Approximations in the Table

Integration tables often use approximations to make calculations simpler and more manageable. These approximations might involve using numerical integration techniques such as the Midpoint Rule, Trapezoidal Rule, or Simpson's Rule. This can lead to a slight difference between the table value and the exact value calculated by a computer algebra system.
02

Different Integration Techniques

A computer algebra system can make use of different integration techniques such as symbolic integration, which finds an exact solution in the form of a function or a symbolic expression. On the other hand, integration tables often provide general family of functions based on certain assumptions, which might not be precise for specific cases.
03

Different Forms of the Integral

The integral of a function can be expressed in different ways, typically as an antiderivative or as a definite integral with limits of integration. The table and computer algebra system might use different forms for representing the same integral, which could ultimately lead to differences in their values.
04

Errors in the Table

It is possible that there are some errors or typos in the integration table, leading to a difference in the values of the integral in question. Human errors may occur during the compiling or updating process of the table.
05

Numerical Instabilities in the Computer System

When evaluating integrals, computer algebra systems might be prone to numerical instabilities, truncation errors, or round-off errors, which can cause a difference between the exact solution and the computer-generated solution. These issues are commonly encountered when dealing with very large or very small numbers or when the integrand has significant fluctuations. In conclusion, the difference in an integral value found in a table and the same integral evaluated by a computer algebra system might arise due to approximations in the table, different integration techniques, different forms of the integral, errors in the table, or numerical instabilities in the computer system.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{e^{x}+e^{2 x}}$$

Use numerical methods or a calculator to approximate the following integrals as closely as possible. $$\int_{0}^{\infty} \ln \left(\frac{e^{x}+1}{e^{x}-1}\right) d x=\frac{\pi^{2}}{4}$$

Evaluate the following integrals. Consider completing the square. $$\int_{2+\sqrt{2}}^{4} \frac{d x}{\sqrt{(x-1)(x-3)}}$$

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{x-\sqrt[4]{x}} ; x=u^{4}$$

For what values of \(p\) does the integral \(\int_{2}^{\infty} \frac{d x}{x \ln ^{p} x}\) exist and what is its value (in terms of \(p\) )?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.