Question

Calculus question!
"5.1: Areas and Distances"

Answer 1

"Estimate the area under the graph of \(g(x)=\frac{5}{1+x^2}\) from \(x=-2\) to \(x=2\) using 4 rectangles at right endpoints."

Answer 2

Riemann sums?
I haven't done these sorts of problems in a long while - but I can try to help ^_^
@kittybasil

Answer 3

Sure, go ahead :)

Answer 4

ehhh
so the idea of riemann sums is to add up rectangles in order to estimate the area under a curve

Answer 5

Yup, got that :)

Answer 6

what they mean by "right endpoints" means that the right corner of these rectangles should be the points touching the given function

Answer 7

Ahh, I see. :O

Answer 8

they want the range of -2 to 2
and they want to estimate using 4 points

thankfully, the way they ask this means that the points end up on whole number values
x = -1, 0, 1, and 2

Answer 9

this means that the base of every rectangle here is = 1

Answer 10

Got it :)

Answer 11

Then to find the area of each of these rectangles - we need the height, right?

how do you think you can get the height of each rectangle? :3

Answer 12

f(x)?

Answer 13

The y-coordinate/output function, I mean.

Answer 14

yup!
you can plug in the x values I listed before to get the respective heights ^_^ that's right :)

Answer 15

and then you just add up all the areas of the rectangles to get the answer ;P

Answer 16

Okay, I think I've got that too.
My main issue is how to put that in Riemann sum form e_e

Answer 17

there's a form?
ehhhh
maybe it's like
answer = (base)*(sum of heights)
= (1) * ( f(-1) + f(0) + f(1) + f(2) )

Answer 18

the base can be factored out because it's the same

Answer 19

Or whatever the \(\Sigma\) is idk

Answer 20

Something like that I think?

Answer 21

is that how that works?

Answer 22

Ooh, okay. Thanks :D

Answer 23

the triangle (delta) x = base

Answer 24

Hmm, the medal system is wonky...

Answer 25

Maybe I need to refresh this page.

Answer 26

yeah, don't worry - I got the medal x'D it just doesn't update properly until you leave the question and come back ;)

Answer 27

Oh lol

Answer 28

the weird E symbol (sigma)
the bottom number is the first number to be plugged in, incrementing at whole numbers until the top number
basically, the weird equation that I drew expands to become (1) * ( f(-1) + f(0) + f(1) + f(2) )
;P

Answer 29

I think it's \Sigma