Can someone give me the solution

Please provide the step by step solution for the integration. Thank You.

\int _{ 0 }^{ 1 }{ \int _{ 0 }^{ 1 }{ |x-y|(6{ x }^{ 2 }y)dxdy } }

|.| = Absolute value.

Note by Ankush Gogoi
3 weeks, 5 days ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

You can get rid of the absolute value by dividing the integration region into two parts: One above the line \( y = x \) and one below it:

\[\large{\int_0^1 \int_0^1 | x - y | (6 x^2 y) \, dx \, dy = \int_0^x \int_0^1 (x - y) (6 x^2 y) \, dx \, dy + \int_x^1 \int_0^1 (y - x) (6 x^2 y) \, dx \, dy } \]

The rest is tedious but trivial

Steven Chase - 3 weeks, 3 days ago

Log in to reply

Thank You @Steven Chase for the approach. I got the answer.

Ankush Gogoi - 3 weeks, 3 days ago

Log in to reply

Would you please post this as a problem in the calculus section?

Steven Chase - 3 weeks, 3 days ago

Log in to reply

@Steven Chase I'm not good at writing using Latex. But I will try to post it.

Ankush Gogoi - 3 weeks, 2 days ago

Log in to reply

@Steven Chase Yeah that also works......but still, the integral is quite long and monotonous to evaluate....!!!

Aaghaz Mahajan - 3 weeks, 3 days ago

Log in to reply

Indeed. I would just integrate it numerically

Steven Chase - 3 weeks, 3 days ago

Log in to reply

@Steven Chase Yeah.but maybe he was practicing multivariable calculus for the first time........then this sort of question works for improving one's understanding......

Aaghaz Mahajan - 3 weeks, 3 days ago

Log in to reply

Is this what you mean? : \(\huge \displaystyle \int_{0}^{1} \int_{0}^{1} |x-y|(6x^2 y) dx \ dy \)

Mohmmad Farhan - 3 weeks, 5 days ago

Log in to reply

Ya. I want the solution for this

Ankush Gogoi - 3 weeks, 5 days ago

Log in to reply

@Aaghaz Mahajan, I have sorted out the \(\LaTeX\). May you help @Ankush Gogoi

Mohmmad Farhan - 3 weeks, 5 days ago

Log in to reply

@Mohmmad Farhan Hey!!! Thanks a lot!!! @Ankush Gogoi Well wait then, I'll send the solution shortly....!!

Aaghaz Mahajan - 3 weeks, 5 days ago

Log in to reply

@Ankush Gogoi Ok......if you are familiar with methods of Multiple Integration, then this is fairly easy by a change of variables.........
Firstly, map x and y to (u+v) and (u-v) respectively and then find the Jacobian.......The motivation behind this step was to remove both variables from inside the modulus operator and replace them with one variable...........Now, after change of variables, apply the desired limits and then open the modulus accordingly.......and thats that!!!! You are done............!!!

Aaghaz Mahajan - 3 weeks, 5 days ago

Log in to reply

Sounds like a good plan... Just a little confused as to the limits of integration once you put in \(x=u+v\) and \(y=u-v\). Kinda feel like I haven't touched this stuff for a good 6-7 years, but I only remember briefly (like the change of variables & the Jacobian associated to it).

Gennady Notowidigdo - 3 weeks, 4 days ago

Log in to reply

Sir, well the limits are inside the square on the u-v plane.......i.e. .......the region is a square with vertices on (0,0),(0.5,0.5),(1,0) and (0.5,-0.5) on the u-v plane.........

Aaghaz Mahajan - 3 weeks, 4 days ago

Log in to reply

@Aaghaz Mahajan Ahk. Got it.

Gennady Notowidigdo - 3 weeks, 4 days ago

Log in to reply

Can you please provide the solution.

Ankush Gogoi - 3 weeks, 5 days ago

Log in to reply

Have you made a decent attempt at the problem? If so, please show. If not, then you should at least try before asking others on this site to give you worked answers.

Gennady Notowidigdo - 3 weeks, 4 days ago

Log in to reply

That is what I have written.....!! What do you need??

Aaghaz Mahajan - 3 weeks, 5 days ago

Log in to reply

@Aaghaz Mahajan I assume he wants you to write out the whole thing for him?

Gennady Notowidigdo - 3 weeks, 4 days ago

Log in to reply

@Gennady Notowidigdo Well.......I really can't and this is the thing I regret........I mean I have ZERO knowledge of programming and LATEX and stuff........and also, I dont have the interest to learn that......so that is why on this site, most of the things that I do are based more on writing....(except solving problems of course!!!).....

Aaghaz Mahajan - 3 weeks, 4 days ago

Log in to reply

@Aaghaz Mahajan It's not that you CAN'T; you SHOULDN'T. I replied to the previous post directly; let's end it on that note.

Gennady Notowidigdo - 3 weeks, 4 days ago

Log in to reply

Well, LATEX is not clear........I don't know what are you asking for.......

Aaghaz Mahajan - 3 weeks, 5 days ago

Log in to reply

Sorry. Not good with latex for writing.

Ankush Gogoi - 3 weeks, 5 days ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...