Download the App!

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

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Represent the given sum (which represents a partition over the indicated interval), by a definite integral, and then evaluate the integral to determine the value of the sum.$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} x_{k}^{2},[1,2]$$

$$\int_{1}^{2} x^{2} d x=7 / 3$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Missouri State University

University of Nottingham

Idaho State University

Boston College

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.

01:21

Represent the given sum (w…

01:54

02:37

Let $P$ be a regular parti…

06:00

Evaluate each limit by int…

01:44

01:08

Definite Integral as the l…

for this problem we are asked to represent the given some is a definite integral and then to evaluate so our some is the limit as N approaches infinity of one over N times the sum from K equals one up to end of x K squared on the interval from 1 to 2. So we can see that what we are actually summing up, our actual function would be x squared. We can see that we're going from 1 to 2. So that indicates to us then that our integral form is going to be the integral from 1 to 2 of x squared dx. So the anti derivative of x squared is going to be x cubed over three. We're now evaluating that from 1 to 2. So that's going to give us two cubed over three -1, cubed over three or -1/3. Uh two cubed is going to be eight. So we have eight minus 1/3. So we have 7/3 as our final result.

View More Answers From This Book

Find Another Textbook

04:04

Evaluate the given integral.$$\int \frac{(\ln x)^{N}}{x} d x$$

01:47

Determine the area of the indicated region.Region bounded by $f(x)=\frac…

02:59

Consider the parabola $f(x)=a x^{2},$ with $a>0 .$ When $x=b,$ call the $…

03:16

Evaluate $\int_{R} \int f(x, y) d A$ for $R$ and $f$ as given.(a) $f(x, …

04:42

Evaluate the given integral.$$\int_{0}^{1} \int_{0}^{2}\left(x^{2}+y^{2}…

09:37

Sketch some of the level curves (contours) or the surface defined by $f$.

02:01

A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)…

03:05

Show that any polynomial $p(x)$ may be written as $p(x)=f(x)+g(x)$, where $f…

06:30

For $f(x, y, z)=2 x^{2}+2 y^{2}+3 z^{2}+2 x y+3 x z+5 y z-2 x+2 y+2 z$ find …

01:43

Evaluate the given definite integral.$$\int_{-2}^{3}|t| d t$$