Change the order of integration?

Question

Here is the equation..sorry i couldn't find a way to paste it
http://img3.freeimagehosting.net/image.php?63beb02fc0.jpg

change the order of integration from dydx to dxdy

Thanks!

Answer 1

The region of integration is the triangle bounded by y = x, y = -x, and x = 2. To integrate w.r.t. x first, the limiting curves change at y=0, so you have to split into two integrals. For -2<=y<=0, you have x from -y to 2, and for 0<=y<=2, you have x from y to 2. The integral with order of integration interchanged is

Int(y from -2 to 0) Int(x from -y to 2) integrand dx dy
+ Int(y from 0 to 2) Int(x from y to 2) integrand dx dy.

Answer 2

I find that I really need to draw a picture of the region of integration to help me invert the order of integration.

You will need to break this into two double integrals. Note that for -2
So for one double integral, (outer integral) y goes from -2 to 0 and (inner integral) x goes from -y to 2; for the other double integral, (outer integral) y goes from 0 to 2 and (inner integral) x goes from y to 2. (And obviously you add these two.)

Answer 3

Just switch the integration limits from the inner and outer with each other and do the "dx" integration first, treating terms in "y" as constants.
The intergrands don't look all that difficult.

Answer 4

?[0,3] ?[x, 3] sin(y)*cos(x/y)/y dy dx changing the order of integration: ?[0,3] ?[0, y] sin(y)*cos(x/y)/y dx dy = ?[0,3] sin(y)/y * y*sin(x/y) eval. from x = 0 to x = y dy = ?sin(y) * (sin(a million) dy = -sin(a million)*[cos(3) - a million] = sin(a million)*(a million - cos(3))