/*! 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 7 Use (a) the Trapezoidal Rule and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

Use (a) the Trapezoidal Rule and (b) Simpson's Rule to approximate the integral. Compare your results with the exact value of the integral. $$ \int_{0}^{1} x e^{=x^{2}} d x ; \quad n=6 $$

Short Answer

Expert verified
Using the Trapezoidal Rule and Simpson's Rule, we have calculated the approximations of the integral \(\int_{0}^{1} x e^{x^{2}} dx\) as follows: \(h=\frac{1}{6}\) Trapezoidal Rule: \(\frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6) \right]\) Simpson's Rule: \(\frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6) \right]\) After evaluating these approximations and comparing them to the exact value of the integral, which is \(\frac{1}{2}(e-1)\), we can analyze the accuracy of each method.

Step by step solution

01

Understand the two methods

In this problem, we will use two methods to approximate the integral: a. Trapezoidal Rule b. Simpson's Rule Both methods use a series of intervals to approximate the area under the curve of the function in the given interval. The different methods use different techniques to provide a closer approximation. Since we are given the number of intervals \(n=6\), we can calculate the value of \(h\) by dividing the total interval length (\(1 - 0 = 1\)) by the number of intervals.
02

Calculate the value of h

Compute the value of \(h\): \[h = \frac{b-a}{n} = \frac{1-0}{6} = \frac{1}{6}\]
03

Create intervals and evaluate the function at interval points

Create intervals by dividing the range \([0, 1]\) into 6 equal parts, and evaluate the function at each interval: \[x_i = a + ih = 0 + i \cdot \frac{1}{6} \Rightarrow x_i = \frac{i}{6} \quad \text{for} \quad i=0,1,2,\ldots,6\] Evaluate the function at each interval: \[f(x_i) = x_i e^{x_i^2}\]
04

Calculate the integral using the Trapezoidal Rule

Use the Trapezoidal Rule formula to approximate the integral: \[\int_{0}^{1} x e^{x^{2}} dx \approx \frac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6) \right]\] Substitute the values of \(h\) and each \(f(x_i)\) and compute the integral approximation.
05

Calculate the integral using Simpson's Rule

Use Simpson's Rule formula to approximate the integral: \[\int_{0}^{1} x e^{x^{2}} dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6) \right]\] Substitute the values of \(h\) and each \(f(x_i)\), and compute the integral approximation.
06

Compute the exact integral value

Calculate the exact value of the integral by finding the antiderivative: The antiderivative for a function like \(x\cdot e^{x^2}\) is not so trivial. To find it, we can use integration by substitution: Let \(u = x^2, du = 2x\,dx\). Then, \(\frac{1}{2}du = x\,dx\), and the new limits are: \(u(0) = 0^2 = 0\) and \(u(1) = 1^2 = 1\). \[\int_{0}^{1} x e^{x^{2}} dx = \frac{1}{2}\int_0^1 e^u du\] Find the antiderivative: \[\frac{1}{2}\int_0^1 e^u du = \frac{1}{2} [e^u]_0^1 = \frac{1}{2}(e^1 - e^0) = \frac{1}{2}(e - 1)\]
07

Compare the approximations with the exact value

Compare the results obtained in Steps 4 and 5 (Trapezoidal Rule and Simpson's Rule approximations) with the exact value obtained in Step 6.

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!

Key Concepts

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

Trapezoidal Rule
The Trapezoidal Rule is a straightforward method for approximating the integral of a function. This rule works by dividing the total interval over which the function is being integrated into smaller sub-intervals of equal length. For each sub-interval, the area under the curve is approximated as a trapezoid rather than fitting the curve perfectly.

Here's how it works:
  • First, divide the interval from \(a\) to \(b\) into \(n\) equal parts. The width of each part, denoted \(h\), is calculated using the formula \(h = \frac{b-a}{n}\).
  • Next, calculate the value of the function at the endpoints of each interval. These values are crucial as they are used in calculating the area of the trapezoids.
  • The Trapezoidal Rule formula for approximating an integral is:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{2}\left[ f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n) \right]\]
One advantage of this rule is its simplicity and ease of use, making it a popular choice for quick approximations. However, its accuracy can sometimes be limited, especially for functions that are not linear, since the rule approximates the curve with straight lines.
Simpson's Rule
Simpson's Rule is another technique to approximate the integral of a function over a given interval. Unlike the Trapezoidal Rule, Simpson's Rule uses parabolic arcs instead of straight lines to approximate the function, providing a more accurate estimate, especially for smooth and continuous functions.

The method proceeds as follows:
  • Divide the interval from \(a\) to \(b\) into an even number of sub-intervals \(n\). Calculate \(h = \frac{b-a}{n}\) as the width of each sub-interval.
  • Calculate the function values at every point. The Simpson's Rule then uses these values to fit a parabolic curve over each pair of intervals.
  • The formula for Simpson鈥檚 Rule is:\[\int_{a}^{b} f(x) \, dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n) \right]\]
This method typically yields better precision with fewer intervals compared to the Trapezoidal Rule. However, it requires the number of intervals to be even and might be more computationally intensive depending on the function being evaluated.
Integral Approximation
Integral approximation is a numerical technique used to estimate the value of definite integrals when finding an exact solution analytically is difficult or impossible. Two common methods for this purpose are the Trapezoidal Rule and Simpson's Rule.

When choosing a method, consider the function:
  • Smooth functions generally yield accurate results with Simpson's Rule due to its use of parabolic approximations.
  • For functions that are not too complex or over small intervals, the Trapezoidal Rule might suffice.
The goal of using numerical methods is to achieve a balance between accuracy and computational efficiency. The accuracy of an integral approximation is often checked by comparing with any available exact solutions, if possible. In practical applications, it's important to know how much error you can tolerate in your approximations.

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

Suppose that a tractor purchased at a price of \(\$ 60,000\) is to be depreciated by the double declining balance method over a 10 -year period. It can be shown that the rate at which the book value will be decreasing is given by $$R(t)=13,388.61 e^{-0.22314 t} \quad 0 \leq t \leq 10$$ dollars per year at year \(t\). Find the amount by which the book value of the tractor will depreciate over the first 5 years of its life.

Find the indefinite integral. $$ \int x^{2}(1-x)^{7} d x $$

Prove that $$ \int_{-1 / 2}^{1 / 2} 2^{\cos x} d x=2 \int_{0}^{1 / 2} 2^{\cos x} d x $$

Special Theory of Relativity According to Einstein's special theory of relativity, the mass of a particle is given by $$ m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} $$ where \(m_{0}\) is the rest mass of the particle, \(v\) is its velocity, and \(c\) is the speed of light. Suppose that a particle starts from rest at \(t=0\) and moves along a straight line under the action of a constant force \(F\). Then, according to Newton's second law of motion, the equation of motion is $$ F=m_{0} \frac{d}{d t}\left(\frac{v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\right) $$ Find the velocity and position functions of the particle. What happens to the velocity of the particle as time goes by?

A horizontal, uniform beam of length \(L\), supported at its ends, bends under its own weight, \(w\) per unit length. The elastic curve of the beam (the shape that it assumes) has equation \(y=f(x)\) satisfying $$ E I y^{\prime \prime}=\frac{w x^{2}}{2}-\frac{w L x}{2} $$ where \(E\) and \(I\) are positive constants that depend on the material and the cross section of the beam. a. Find an equation of the elastic curve. Hint: \(y=0\) at \(x=0\) and at \(x=L\). b. Show that the maximum deflection of the beam occurs at \(x=L / 2\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.