#estimating riemann sum

169 messages · Page 1 of 1 (latest)

lone kettle
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Did i do this right ?

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static valleyBOT
lone kettle
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My attempt

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Is there a formula im supposed to follow?

neon sparrow
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Why didnt you use

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f(x) = -0.5x² + 8

burnt fernBOT
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𝔸dωn𝓲²s

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neon sparrow
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6.5 and 5.5 are way too off

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And 4.5 especially

lone kettle
lone kettle
neon sparrow
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it's literally

lone kettle
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i might have done it incorrectly

neon sparrow
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f(1) + f(2) + f(3) + f(4)

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Δx = 1 doesnt matter here

neon sparrow
lone kettle
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okok, let me check my work rq

neon sparrow
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okok

lone kettle
neon sparrow
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No

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I am saying multiplying things by 1 is trivial

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So you might aswell spare yourself writing times 1 everytime

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Also notice 24

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cannot be

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the actual integral is 21.333 approximately

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you are drawing rectangles inside

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meaning you will even miss some area

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and not have extra area

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So your result is even supposed to be acshually less than 21.333

neon sparrow
# burnt fern **𝔸dωn𝓲²s**

This is just a short way of saying I am summing up rectangles (here 4) with width Delta x (here 1) and height the function value at right corner

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Sigma means sum

lone kettle
neon sparrow
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ye

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yea yea 🥱

lone kettle
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What to do with left side riemann sum?

neon sparrow
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idk

lone kettle
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Do I subtract

neon sparrow
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Left side points?

lone kettle
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Mhm

neon sparrow
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I am bad at English

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and Math

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so can you be a little more clear?

lone kettle
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Like this one

neon sparrow
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see

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left side endpoints

lone kettle
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yeah!

neon sparrow
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aschually 🤓

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you are missing one rectangle

lone kettle
neon sparrow
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0 to 1?

lone kettle
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U could barely see it

neon sparrow
neon sparrow
lone kettle
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I drew it 😭

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back to the question, mathematician

neon sparrow
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where 🤓?

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neon sparrow
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i play fortnite

mental hollow
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math bullyin is crazy

lone kettle
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Do i use the same concept as the right one and apply with the left but subtract it instead?

lone kettle
neon sparrow
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why subtract?

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you are adding

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rectangles again

neon sparrow
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This time you will have some extra area

lone kettle
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Whats the difference between left and right?

neon sparrow
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so you would expect something greater but still close to 21.3

neon sparrow
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missing out on area

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left is as you may see

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the rectangles are outside the curve

mental hollow
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u nearly studyin integrals?

neon sparrow
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so you would have extra area instead left

mental hollow
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snooze

neon sparrow
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and integral is like in the middle of these two calculating the exact result

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exact area

lone kettle
mental hollow
lone kettle
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Yes

mental hollow
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calm

lone kettle
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not calm

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im burnt out

neon sparrow
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take a break

lone kettle
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Its ok, I understand quick

neon sparrow
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good

lone kettle
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The difference is just the rectangle inside and out?

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but calculations are the sa@e

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same

neon sparrow
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ye

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the procedure is the same

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but the result isn't as you may tell graphically

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Generally this is the formula $\sum_{i=0}^n f(x_i) \cdot \Delta x$ which kinda resembles $\int_a^b f(x) : \dd x$.

burnt fernBOT
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𝔸dωn𝓲²s

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neon sparrow
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x_i is just the x values

lone kettle
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Ok next question

neon sparrow
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Ok 🫡

lone kettle
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What about midpoint rectangle?

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Wait i think i drew it wrong

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There should only be two triangles right ?

neon sparrow
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triangles?

lone kettle
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RECTANGLES

neon sparrow
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ok

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they want you two rectangles

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n = 2

lone kettle
neon sparrow
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yes!

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I was about to draw it

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key is to hit the middle

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this is a more compensating approach

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you have area left

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but you are also missing out on area

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so they roughly cancel each other out

lone kettle
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Is there a formula i could follow to show my work

neon sparrow
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It's basically the same approach

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see my red points

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aka the middle

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these are your function values

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because that's your rectangles height

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so f(1) and f(3)

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width you did Delta x = 2 which is correct

burnt fernBOT
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𝔸dωn𝓲²s

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neon sparrow
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I factorized Delta x

burnt fernBOT
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𝔸dωn𝓲²s

587715738484211754.png
lone kettle
neon sparrow
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should add up yea

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so there is a bit area left

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but still a good approximation

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since 22 > 21.333

lone kettle
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What about a trapezoid?

neon sparrow
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We are now summing up trapezoidal areas

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So instead width x height

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I think it was

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$\sum_{i=0}^n \frac{1}{2}(f(x_i)+f(x_{i+1})) \cdot \Delta x$

burnt fernBOT
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𝔸dωn𝓲²s

587715738484211754.png
lone kettle
neon sparrow
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I derived it from my head

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neon sparrow
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where a and b are f(x_i) and f(x_i+1)

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and h Delta X

lone kettle
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What is xi and xi +1

neon sparrow
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f(x_i) is the previous functions value

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f(x_i+1) is the next

neon sparrow
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f(x_0) would be f(0)
f(x_0+1) = f(x_1) = f(1)

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like it's the next function value after a Delta x step

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So your first trapezoid would be (f(0) + f(1)) * 1/2

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ok I derived it correctly

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I hope this clears up

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the x_i and x_i+1

lone kettle
neon sparrow
lone kettle
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OH

neon sparrow
lone kettle
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silly me

neon sparrow
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it's one step iteration

lone kettle
lone kettle
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.solved

static valleyBOT
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Solved

Post marked as solved by @lone kettle.

Use .unsolved if this was a mistake.

neon sparrow