Question

\[\int_0^1 \int_0^1 xy \sqrt{x^2 + y^2} \text{dydx}\]

i get confused with multiple variables :/

Answer 1

Could you use polar cordenates?

Answer 2

dont get confused for this we look at in what we are going to respect this function by so in this case we are integrating this function in respect to y so just basically treat EVERYTHING as a constant excpet for y if that still confuses you just put a number as in for x and just sub it later

so to do this we need to do it by subsitution so let u =x^2 +y^2 and du = 2ydy and now (1/2)xdu=xydy and now we just sub this in to get (1/2)x*sqrt(u)du and we easily know how to integrate this which is (1/2)x*(2/3)u^(3/2) = (1/3)xu^3/2 and now we sub it back to get (1/3)x(x^2+y^2)^(3/2) from 0 to 1 and we sub this in as y coordinates

Answer 3

to get (1/3)x(x^2+1)^(3/2) - (1/3)x(x^2)^(3/2) and now we integrate this in respect to x from 0 to 1 woudl you be able to carry on and are there are any questions ?

Answer 4

what do you all mean by polar coordinates?

Answer 5

i guess i understand @hamza_b23 's solution somehow...

Answer 6

ugh..i dont think we've learned that yet :/

Answer 7

No problem, it is not necesary. But I did an exam today and we used that a lot. xD

Answer 8

well ... the one way would be to evaluate it keeping one constant.

Answer 9

yeah..that's where i always get confused :/

Answer 10

Is like doing parcial derivatives...think the other variable as a number.

Answer 11

it's confusing @_@ the partial derivatives i got though :/

Answer 12

mmhmm

Answer 13

uhmm there was that trick where you can switch dx and dy right? how does that go again? it's buried somewhere in my other questions -_-

Answer 14

the one i asked one month ago -_- totally buried...ill try looking for it

Answer 15

yeah you have to chanfge the order of integration to actually sovle the problem properly for example

Answer 16

like for that you have to change the order of integration right ill just leave this posted if you can do this then great if not i will help you through

Answer 17

But, in the integral that he wrote, changing the order of integration doesn´t help. Because you have "x.y" outside the sqrt. So is the same to start integring across y, or across x.
But yes, sometimes it really simplify the work.

Answer 18

o no th eintegral he wrote was wawayy simpler than what i posted you should have been able to figure out that was direct integration from what he wrote like if you do mine just directly you will get stuck UNLESS you change the order of integration right