Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

Express the limit as a definite integral on the given interval.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n x_i \sqrt{1 + x_i^3} \, \Delta x $, [2, 5]

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by Yuki Hotta

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

00:27

Frank Lin

00:20

Amrita Bhasin

01:01

Stephen Hobbs

Calculus 1 / AB

Chapter 5

Integrals

Section 2

The Definite Integral

Integration

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

0:00

Express the limit as a de…

00:59

Express the limit as a def…

02:42

00:19

07:25

01:19

Express the given limit as…

00:26

02:28

Express the following limi…

00:16

All right, let's go ahead and do this problem. So the question ISS you're given the limit as an approaches to infinity. The some from Aiko's toe want to end off XVIII cleared of one plus x of Ike Uche Delta X okay and X is between two and five inclusive. What is the definite integral form off this expression? So the definite integral, the integral from a to B of ffx the the X this is defined to be the limit of a remonstrance. Some limit as angles to infinity off the some off I equals toe Want to end off f off X I using the same notation Delta X, where Delta X is equal to B minus a over and where the X is defined between. We need to be okay. So long story short. What's really happening here is that you have a region between a to B, and then you want to calculate the area under the curve using rectangles where this is the height and this is the width. Graphically speaking, this is what's going on. So it just doesn't example if I have a shape that looks like this. This is a this is B. I split it into a bunch of rectangles where this is f of X, Delta X is each of these width in the height is f off X I where the X I is each of these points. Okay, so we know that the area under the curve right here the approximation gets better and better the more rectangles we have. So if we use in rectangles, this is going to be the sum. But as we use the limit as an approaches to infinity, we're gonna have an exact match. So this is the definition off what the definite integral is. Okay, so all we need to do to really answer this question is to see how this is related to that expression right here. It all boils down into figuring out what the limit off the integration is. What effort X is, and de avec dx is usually just Delta X. So we don't need to worry about that part. So here the limit and the some we already know that part is just going to be this s portion of the integral. Now here, X, I is my input. We need to match this as my ex, I What is illegal too? Well, it's X I times the square root of one plus X I Q. Actually, not that difficult. It's exactly what it looks like. This portion becomes your Delta X. So when I multiply f of X, I with Delta X, I get this portion, which is the area under the curve. That's a rectangle. Okay, if we add them all up, it will be the definite integral. And we're taking the limit as and goes to infinity. So we know that it is going to be the exact answer. Now what is the what are the limits off the integration it's given right here? X is supposed to be from 2 to 5, so that is where we get the expression. The integration from 2 to 5 f of x is X square. Root off one plus x you and the exchanges to DX. And this is going to be the answer to this question.

View More Answers From This Book

Find Another Textbook

00:52

A survey of random sample of 210 male teens and 228 female teens. ages 13 ye…

03:43

(10 marks) Drawing Straws= Four soldiers have to decide between themselves, …

02:48

! [ 1 J/s ~ln 1 { ! E ui~ 8 3 2 [ 88 Nls 1 1 L 1 } ! { J8 ] 10 9 5 …

01:18

Determine whether the quantitative variable is discrete or continuous_Ti…

00:47

9 Find the exact value.tan 25 +tan 201-tan 25 tan 20

05:28

An equation of = hyperbola is given.3y_x_9=0 Find the vertices, foci, an…

03:41

The engine in airplane has power curve approximated by 221 y = 5000 where I …

02:26

Q1 The length of time between breakdowns of an essential piece of equipment …

07:05

02 In a chemical processing plant it is important that the yield of a certai…