/*! 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 2 Explain geometrically how the Mi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 2

Explain geometrically how the Midpoint Rule is used to approximate a definite integral.

Short Answer

Expert verified
In summary, the Midpoint Rule is a geometric technique used to approximate the area under a curve (represented by a definite integral) by using the midpoints of sub-intervals created by partitioning the given interval. It computes the area by summing up the areas of rectangles with heights determined by the function's value at the midpoint of each sub-interval and with widths equal to the width of each partition. The accuracy of the approximation improves as the number of partitions (n) increases.

Step by step solution

01

Understanding the definite integral

A definite integral represents the area under a curve for a specific interval. Mathematically, it is defined as: \[ \int_a^b f(x) dx \] where f(x) is the function being integrated, and 'a' and 'b' are the lower and upper limits of the interval.
02

Partitioning the interval of integration

To approximate the area under the curve, we can partition the interval `[a, b]` into 'n' equal sub-intervals. Each of these sub-intervals will have a width '螖x' where: \[ 螖x = \frac{b - a}{n} \] These partitions will help create rectangles, whose area will sum up to approximate the area under the curve.
03

Defining the Midpoint Rule and its geometric interpretation

The Midpoint Rule approximates the definite integral by using the midpoints of each partition to compute the area of the rectangles. The heights of the rectangles are determined by the value of the function at the midpoint of the sub-interval, represented as: \[ M_n = 螖x \left[ f \left( \frac{a + a + 螖x}{2} \right) + f \left( \frac{a + 螖x + a + 2螖x}{2} \right) + \cdots + f \left( \frac{a + (n-1)螖x + a + n螖x}{2} \right)\right] \] Geometrically, the Midpoint Rule uses the midpoints of the sub-intervals to create rectangles whose total area is an approximation of the area under the curve. The accuracy of the approximation improves as the number of partitions 'n' increases.

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

Solve the following problems using the method of your choice. $$\frac{d z}{d x}=\frac{z^{2}}{1+x^{2}}, z(0)=\frac{1}{6}$$

Determine whether the following statements are true and give an explanation or counterexample. a. The general solution of \(y^{\prime}(t)=20 y\) is \(y=e^{20 t}\). b. The functions \(y=2 e^{-2 t}\) and \(y=10 e^{-2 t}\) do not both satisfy the differential equation \(y^{\prime}+2 y=0\). c. The equation \(y^{\prime}(t)=t y+2 y+2 t+4\) is not separable. d. A solution of \(y^{\prime}(t)=2 \sqrt{y}\) is \(y=(t+1)^{2}\).

Use the reduction formulas in a table of integrals to evaluate the following integrals. $$\int \sec ^{4} 4 t d t$$

Use integration by parts to evaluate the following integrals. $$\int_{0}^{\infty} x e^{-x} d x$$

An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), \(f(x)=e^{-\alpha x^{2}}\). a. Graph the Gaussian for \(a=0.5,1,\) and 2 b. Given that \(\int_{-\infty}^{\infty} e^{-a x^{2}} d x=\sqrt{\frac{\pi}{a}},\) compute the area under the curves in part (a). c. Complete the square to evaluate \(\int_{-\infty}^{\infty} e^{-\left(a x^{2}+b x+c\right)} d x,\) where \(a>0, b,\) and \(c\) are real numbers.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

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.