Sketch the region of integration, reverse the order of integration, and evaluate the integral

- UsukiDoll

Sketch the region of integration, reverse the order of integration, and evaluate the integral

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- katieb

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- UsukiDoll

##### 1 Attachment

- UsukiDoll

this is what I got so far...how do I evaluate this in terms of x when there is a y inside the sin?
@wio

- UsukiDoll

Moreover is there a simpler way to find the new values for reversing the integral?

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## More answers

- UsukiDoll

I feel like I'm treading into unknown waters. except for the dydx ones...since it's type 1 and I have to go from left to right to get type II. It seems that the y values are there when I look in the same direction as bottom to top

- UsukiDoll

@hartnn ???

- UsukiDoll

what I did is in the attachment. It's easier to do it that way rather than drawing it on here

- UsukiDoll

http://assets.openstudy.com/updates/attachments/522e9a27e4b0fbf34d63d88f-usukidoll-1378785914685-14.226.png

- hartnn

what you did is correct....
now first we will be integrating w.r.t x
right ?
so in sin (xy), 'y' will be treated as constant
same as integrating sin (ax)

- UsukiDoll

holy crud I actually got the new integral values correct?

- UsukiDoll

is there any easier way to remember or a tip or something? It's just that part that drives me nuts.

- UsukiDoll

that would be...-cos(ax)

- hartnn

to remember ?
nothing to remember....
first you find out the region, here it was,
|dw:1378786640140:dw|
right ?

- UsukiDoll

yeah

- hartnn

so, previously it was y=x to y=2 (vertical lines)
to change the order we just make horizontal lines in same region
|dw:1378786772700:dw|
that would be x=0 to x=y
as you correctly mentioned :)

- hartnn

now since we are FIRST integrating w.r.t x
we will treat y as constant
so integral of sin (xy) will be -cos (xy)/y
don't forget to divide by constant (here y)
got this ?

- UsukiDoll

ok now I go tit :) so then it would be F(y)-f(0)

- hartnn

yup, go ahead

- UsukiDoll

-cos(yy)/y - [-cos(0)/y]

- UsukiDoll

|dw:1378786967712:dw|

- UsukiDoll

|dw:1378787010839:dw|

- UsukiDoll

|dw:1378787049911:dw|

- UsukiDoll

and then integrate with respect to y and then F(2)-F(0)

- hartnn

don't forget the 2y^2 which was there before, which you took out of the integral w.r.t x, because it was a constant...remember ?

- UsukiDoll

oh yeah so ...|dw:1378787239080:dw|

- hartnn

|dw:1378787294972:dw|

- hartnn

and 2y^2 will get distribute to both terms

- UsukiDoll

|dw:1378787346996:dw|

- hartnn

yeah, note the 'y' s getting cancelled...
then that an easy integral to integrate

- UsukiDoll

|dw:1378787460966:dw|

- UsukiDoll

u=y^2 du = 2y

- UsukiDoll

???

- hartnn

good,
go ahead

- hartnn

u= y^2
du = 2ydy

- UsukiDoll

|dw:1378787668981:dw|

- hartnn

corret

- hartnn

but don't forget to change the LIMITS of integration

- UsukiDoll

|dw:1378787745703:dw|

- UsukiDoll

F(2)-F(0)

- hartnn

you can resubstitute back...but i think it will be beter if you had changed the limits, when you did u=y^2

- UsukiDoll

nah I rather resubsitute

- hartnn

ok, so u getting 4-sin4....do you have answer ?
is this correct ?

- UsukiDoll

-sin4+4 ???? what the

- UsukiDoll

yup :)

- UsukiDoll

gawd dam**** I must selfesteem issues with the subject or something

- hartnn

correct? so i need not go through it again ?

- UsukiDoll

huh?

- UsukiDoll

I got -sin4+4

- hartnn

i mean i was going through it again, just to make sure all steps were correct...if you had an answer, then there was no need of this

- UsukiDoll

so it's -sin4+4?

- UsukiDoll

o_O

- hartnn

i am checking steps...

- hartnn

yes! -sin 4 +4 is correct :)

- UsukiDoll

yay!!! and i do have self esteem issues with this subject like what the heck I used to fly right through this bt i now i break down every time I solve a problem

- UsukiDoll

:/ plus all the old material is coming back. argh that's what happens when I don't use any calculus material for a semester

- hartnn

good luck :)

- UsukiDoll

thanks :)

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