Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Using a Riemann Sum Determine$$\lim _{n \rightarrow \infty} \frac{1}{n^{3}}\left(1^{2}+2^{2}+3^{2}+\cdots+n^{2}\right)$$by using an appropriate Riemann sum.

$\frac{1}{3}$

Calculus 1 / AB

Chapter 4

Integration

Section 3

Riemann Sums and Definite Integrals

Integrals

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

00:43

Evaluate the limit by firs…

02:51

00:38

06:00

Evaluate each limit by int…

00:08

Express the limit as a def…

03:40

07:41

Riemann sums to integrals …

03:30

Compute the sum and the li…

02:46

03:46

So when you use an appropriate Riemann some toe fight with this 11 over and cute plus one squared plus two squared, uh, post three squared buzz and squared. So, uh, you know, the function is you can see that is ah, it's miserable Squares right. One squared, plus two squared. Plus this word so we can let every thanks b x squared. And then we have limit between zero and one. We're trying to determine the appropriate remind some for this one. Then, uh, you know, removed. Cem is, uh, summation. I from one toe end F o c. I write change in X I in your necks. So, uh, what is the change in X change in X? It's how it should be minus a over end. And so, what is our be Arby's one, right? And our eight zeros. So it's just one of red. And what is C I c. I is a plus changing X. I write a plus change in X I a 00 close. So we're just gonna write changing eggs, genuine exits, one over end. And then I saw the Riemann some. It's just gonna be summation. I from one toe end f off egg. Ever see? I see eyes now, uh, one of her and I rises one over and I times changing ex change in X is just one over. And so what is one ever off one over? And I means I will write c s here. I'm gonna put one over, and I So this is just estimation I from one toe end one and I squared one over in, right? And what is that? That is summation. I from one toe end. I squared right over n squared times, one over end. And that finally is gonna be this, right? That one on a different page. So summation I from one toe end, f c I changing X is so fine that this one is gonna be summation. I'm from one to end, right. I squared R and cubic because this one is one of the end times One of route times out and squared one of her and square. That is one of her in cubic and us in here. I could bring it out, right? I could bring it out. So when I bring it out that I have won over and cubic in summation, I won, and then I squid. So you see that this is exactly the same as the 1st 1 here. Can you see that? This one of her end cubic. Is this one over in Cuba here. And then this whole thing here, everything here in the parentheses is the summation of ice. Where right, so that is the appropriate relearn some. So if you have this one, this is gonna be one over n cubic. Now, what is one of her I squared is just n n plus one to end plus one over six, right? Yeah. Now that is Jezzie re months. Um, Now we have to find the limit as an approaches infinity of the three months, Um, so the limit as an approaches infinity of the Riemann. So, um, which is F o c I that It's just the limit as an approaches Infinity of one and cubic right. And close. One two U. N. Plus one over six. You see that? So we have this one, then, uh, this end here is gonna take one of these. So this is gonna be this is gonna be a limit and approaches Infinity. Ah, endless one to U N plus one over six. And squid now have toe expand this read this current disease. So what is expansion of this one in the red might limit here? First expansion of this one is gonna be to end squared, right? Because no, this is three. And and then close one that is expansion. And then over six. And suede. This is the same as now. We're gonna just, uh, distribute the denominator to all the numerator. So this is gonna be a limit and approaches Infinity now to n squared, divided by six n squared. That's just one of the three, right? Then purse three and divided by six. And swear that it's just one over to end, then plus one over six and swear. So the limit as an approaches infinity of this one, anything that has and a denominator is gonna go to zero. So finally, what you're gonna have is just one over three. So that is the limit won over three. So one over three. Okay? I cannot Yeah,

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Evaluate the limit by first recognizing the sum as a Riemann sum for a funct…

Evaluate each limit by interpreting it as a Riemann sum in which the given i…

Express the limit as a definite integral.$$\lim _{n \rightarrow \infty} …

Riemann sums to integrals Show that $L=\lim _{n \rightarrow \infty}\left(\fr…

Compute the sum and the limit of the sum as $n \rightarrow \infty.$$$\su…

01:34

Particle Motion Repeat Exercise 103 for the positionfunction given by

00:40

In Exercises $33-54,$ find the derivative of the function.$$y=5 e^{x^{2}…

03:13

In Exercises $75-80,$ use logarithmic differentiation to find $d y / d x .$<…

02:33

Evaluating an Expression In Exercises $25-28$ , evaluate each expression wit…

11:03

Using Symmetry Use the symmetry of the graphs of the sine and cosine functio…

03:24

Determining Differentiability In Exercises $85-88$ , find the derivatives fr…

00:25

In Exercises $41-56,$ find the derivative of the function.$$f(t)=\operat…

10:06

Area Find the area of the largest rectangle that can be inscribed under the …

01:53

In Exercises $33-54,$ find the derivative of the function.$$y=e^{x-4}$$<…

00:42

In Exercises $15-20$ , sketch the graph of the function.$y=4^{x-1}$$

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.