Question

Sketch the region of the following double integral.
Then change the order of integration and evaluate.
Explain the simplification achieved by interchanging the order.

S=integral
S(0to1)S(0 to pi/2) 6x cos(2xy)dx dy

Answer 1

I am a little confused by the question I emailed my professor he said it should come out to a number which doesnt make sense to me form the question

Answer 2

\[\int\limits_{1}^{0}\int\limits_{\pi/2}^{0} 6x \cos(2xy)dxdy\]

Answer 3

limits need to be switched :)

Answer 4

haha whoops typo :)

Answer 5

h/o let me give it a try

Answer 6

thankyou!

Answer 7

Ok so the way I see it is this:
When you try to take the double integral with respect to x and then wrt y, it is quite difficult to do by hand. You need to do integration by parts, but you end up with extra y's.
If you interchange the bounds, it is easier to go wrt y first, because then you isolate the x's and can then do clean integration by parts.
The answer is 3, btw.

Answer 8

okay I got a ton of garbage when I did it out figured it was wrong, let me trying doing Y first! btw your answer was correct :)

Answer 9

To show you what I mean,
if you integrate wrt x first, you get
\[(3(\cos(\pi*y)+\pi*y*\sin(\pi*y)-1)/(2y^2)\]
which is a nightmare to integrate by hand.

If you integrate wrt y first, you get
\[3\sin(2*x)\]

Answer 10

Which is sooooo easy to integrate by hand :)

Answer 11

holy! haha I never would of thought of that

Answer 12

And sorry I wasn't able to answer your other question about the lagrange multipliers I totally forgot those

Answer 13

I wish they would of just put that in the first place haha, That makes it very dooable and I got 3 as well thanks!

Answer 14

no worries I got a good example to follow. Other wise its just a huge problem to do out and I kept getting lost :).

Answer 15

:) yeah it's a weird trick that some guy thought up
I don't remember what the rule is called, but in the future if you come to some double integrals that you can't solve, you may need to change variables !

Answer 16

Fubini's Theorem

Answer 17

cool, thanks cruffo for the reference!