how to do this. i know that integration will be for rational is x^2/2, irrational will be x^3/3

[slate flint]

x is rational and x^2 when irrational so √ 3 is irrational in the interval and options are wrong?

3/2+8/ 3=25/6

[jaunty slate]

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[simple bobcat]

your reasoning?

$\abs{\int_{[0,2]}f(x)\diff x-\int_{[0,2]}x^2\diff x}\leq \sum_{n=1}^{\infty}M\text{Vol}(r_n-2^{-n+\beta}, r_n+2^{-n+\beta})$

[spice berry]

Cofailure

[simple bobcat]

for all $\beta\in\bN$

[spice berry]

Cofailure

[simple bobcat]

and large enough M

This is possible due to the rationals being countable and hence too scarce to make a difference in a sense as compared to the rest of [0,2]

[simple bobcat]

this is $\leq M\cdot \mu([0,2]\cap\bQ)=M\cdot 0$ as $\beta\to 0$

where mu is the lebesgue outer measure, in this case 0

[spice berry]

Cofailure

[simple bobcat]

Here $\text{Vol}(L)=b-a$ for open intervals $L=(a, b)\subset\bR$

[spice berry]

Cofailure

[slate flint]

Broooooo

Explaining things in mathematics

[simple bobcat]

what do you mean?

[slate flint]

Alg alg integration kar liya

[simple bobcat]

is there something wrong in my process? It has been a long time since I did this so let me know

[slate flint]

√2,√3 irrational

[simple bobcat]

Matlab?

[simple bobcat]

what

[slate flint]

Process m wrong ni h

[simple bobcat]

alg alg bhi kar sakte ho

[slate flint]

Lekin bhai ap jaise explain karte vo begginer k liye sahi thodi rahega

[simple bobcat]

ek hi scene he

[slate flint]

Utna level ka maths aati toh question ni puchte

[simple bobcat]

yes it'll suffice for anyone who has learnt riemann integration

it's your fault for not studying definitions

[slate flint]

I just read again lower and upper riemann definition

[simple bobcat]

If $\mu(E)=0$ then $\int_{X\setminus E} f=\int_{X} f $

[spice berry]

Cofailure

[simple bobcat]

Measures aren't really needed here but it makes explaining it easier

[simple bobcat]

this is the main idea at the end here

specially for countable E

[slate flint]

I will killll uuuu

😂

[simple bobcat]

we can remove smaller and smaller intervals centered around the element of E to show than E has negligible effect

here's another example

[slate flint]

Question ki g*** mat maro

Making doubts into doubts

[simple bobcat]

what do you not understand in the above process?

[simple bobcat]

this one

[slate flint]

Absolutely

alpha+beta and M volume?

hum toh base × height karte the bas

Area of small rectangle

[simple bobcat]

I explained it later on scroll down

Also they ask for the upper riemann integral

@slate flint are you there?

[simple bobcat]

In the upper riemann integral, U(f) = inf U(f, P), in the summation for U(f, P)s we are taking the supremum of the function over the intervals in said partitions P and as they become finer and finer we can use the fact than x>=x^2 on [0,1] and x=<x^2 on [1,2] to get the upper integral

not very rigorous but this points towards ||D|| as the answer

[slate flint]

I was looking for youtube

[slate flint]

In the partition the irrational and rational will be seperated?

[simple bobcat]

use the definition of the upper Riemann integral with the fact than rationals are dense in R

[slate flint]

1 is supermum in first interval

4 in the second

[simple bobcat]

that's over the whole interval

we work in arbitrarily small ones when evaluating U(f)

[cedar nymph]

how is f defined?

[slate flint]

HERE f(x)=x when x is rational and x^2 when x is irrational

[cedar nymph]

divide [0,2] into 2n pieces

and find the upper sum

[slate flint]

@cedar nymph

@simple bobcat

[simple bobcat]

yes

[slate flint]

+close