Question

Desperately need help with this calculus question!

Answer 1

The hint was that dx equals 4/n and x_i equals 4i/n

Answer 2

Do you have this so far?
\[\Large f(x) = \frac{x^2}{4} + 6\]

\[\Large f\left(x_i\right) = \frac{\left(x_i\right)^2}{4} + 6\]

\[\Large f\left(x_i\right) = \frac{\left(\frac{4i}{n}\right)^2}{4} + 6\]

Answer 3

Yes!

Answer 4

ok what do you get when you simplify that?

Answer 5

\[\left( 4*i \right)^{2}/n ^{2}+6\]

Answer 6

you forgot about the 4 down below

Answer 7

Here is what I get
\[\Large f(x) = \frac{x^2}{4} + 6\]

\[\Large f\left(x_i\right) = \frac{\left(x_i\right)^2}{4} + 6\]

\[\Large f\left(x_i\right) = \frac{\left(\frac{4i}{n}\right)^2}{4} + 6\]

\[\Large f\left(x_i\right) = \frac{\frac{16i^2}{n^2}}{4} + 6\]

\[\Large f\left(x_i\right) = \frac{16i^2}{4n^2} + 6\]
Hopefully you agree?

Answer 8

Yup, I cancelled out the 4 in the denominator

Answer 9

oh right, then you'd have

\[\Large f\left(x_i\right) = \frac{4i^2}{n^2} + 6\]

Answer 10

\[\Large R_n = \sum_{i=1}^{n}f\left(x_i\right)\Delta x\]

\[\Large R_n = \sum_{i=1}^{n}\left(\frac{4i^2}{n^2} + 6\right)*\frac{4}{n}\]

\[\Large R_n = \sum_{i=1}^{n}\left(\frac{4i^2}{n^2}*\frac{4}{n} + 6*\frac{4}{n}\right)\]

\[\Large R_n = \sum_{i=1}^{n}\left(\frac{4i^2}{n^2}*\frac{4}{n}\right) + \sum_{i=1}^{n}\left(6*\frac{4}{n}\right)\]

\[\Large R_n = \sum_{i=1}^{n}\left(\frac{16i^2}{n^3}\right) + \sum_{i=1}^{n}\left(\frac{24}{n}\right)\]

\[\Large R_n = \frac{16}{n^3}\sum_{i=1}^{n}\left(i^2\right) + \frac{24}{n}*\sum_{i=1}^{n}\left(1\right)\]

Answer 11

now you'll use the identities

\[\Large \sum_{i = 1}^{n}(i^2) = \frac{n(n+1)(2n+1)}{6}\]

\[\Large \sum_{i = 1}^{n}(1) = n\]

Answer 12

So then you just plug the identities into your last equation for R_n?

Answer 13

yes then you simplify

Answer 14

I got this \[\left( 32*n ^{2}+48*n+16 \right)/(6*n^2)\]

Answer 15

this is what I'm getting (see the attached PDF)

Answer 16

if it's too small, zoom in to at least 150%

Answer 17

What the bah-jesus. You're right. Sorry, we didn't do anything like this in class and I'm incredibly confused

Answer 18

so you're probably a lesson or two ahead?

Answer 19

where are you stuck at?

Answer 20

what I actually put for Rn and how I know when the work is done haha

Answer 21

there is a number of different possible ways to enter Rn but one thing that I would type in, if I were doing the HW is to type in what you see on the last line (since it's the most simplified)

Answer 22

Yes, that's correct! Thank you so much! How would you go about the limit as n goes to infinity?

Answer 23

Actually, I figured it out! Thanks so much for your help!