/*! 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 Suppose you want to approximate ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 3

Suppose you want to approximate the area of the region bounded by the graph of \(f(x)=\cos x\) and the \(x\) -axis between \(x=0\) and \(x=\pi / 2 .\) Explain a possible strategy.

Short Answer

Expert verified
Answer: The right Riemann sum for this approximation is given by \(\sum_{i=1}^n \cos\left(\frac{i\pi}{2n}\right) \cdot \frac{\pi}{2n}\), where \(n\) is the number of subintervals.

Step by step solution

01

Determine the interval and number of subintervals

In this exercise, the interval is \(x \in [0, \frac{\pi}{2}]\). Let's say we want to divide this interval into \(n\) subintervals, each of equal width.
02

Calculate the width of each subinterval

To calculate the width of each subinterval, we need to divide the length of the interval by the number of subintervals \(n\). The length of the interval is \(\frac{\pi}{2} - 0 = \frac{\pi}{2}\). So, the width of each subinterval will be \(\Delta x = \frac{\frac{\pi}{2}}{n} = \frac{\pi}{2n}\).
03

Choose the representative points

We'll use right Riemann sum for our approximation. So, we'll choose the right endpoint of each subinterval as the representative point. For the \(i^{th}\) subinterval, the representative point will be \(x_i = 0 + i\Delta x = 0 + i\frac{\pi}{2n} = \frac{i\pi}{2n}\), where \(i = 1, 2, ..., n\).
04

Calculate the Riemann sum

The right Riemann sum is given by: \(\sum_{i=1}^n f(x_i)\Delta x\) Substitute the values of \(f(x_i)\) and \(\Delta x\): \(\sum_{i=1}^n \cos\left(\frac{i\pi}{2n}\right) \cdot \frac{\pi}{2n}\) Now, evaluate this sum to approximate the area under the curve between \(0\) and \(\frac{\pi}{2}\). To improve the accuracy of the approximation, increase the value of \(n\) (i.e., the number of subintervals).
05

Interpret the result

The result obtained from the Riemann sum is an approximation of the area of the region bounded by the graph of \(f(x) = \cos x\) and the \(x\)-axis between \(0\) and \(\frac{\pi}{2}\). By increasing the number of subintervals, we can improve the accuracy of our approximation.

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

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$

General results Evaluate the following integrals in which the function \(f\) is unspecified. Note \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f\). Assume \(f\) and its derivatives are continuous for all real numbers. \(\int\left(f^{(p)}(x)\right)^{n} f^{(p+1)}(x) d x,\) where \(p\) is a positive integer, \(n \neq-1\)

Periodic motion An object moves in one dimension with a velocity in \(\mathrm{m} / \mathrm{s}\) given by \(v(t)=8 \cos (\pi t / 6)\) a. Graph the velocity function. b. The position of the object is given by \(s(t)=\int_{0}^{t} v(y) d y,\) for \(t \geq 0 .\) Find the position function, for \(t \geq 0\) c. What is the period of the motion - that is, starting at any point, how long does it take the object to return to that position?

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=\left(1-x^{2}\right)^{-1 / 2} ;[-1 / 2, \sqrt{3} / 2]$$

Substitutions Suppose that \(p\) is a nonzero real number and \(f\) is an odd integrable function with \(\int_{0}^{1} f(x) d x=\pi\) a. Evaluate \(\int_{0}^{\pi /(2 p)} \cos p x f(\sin p x) d x\) b. Evaluate \(\int_{-\pi / 2}^{\pi / 2} \cos x f(\sin x) d x\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.