Question

\[f(x,y)= \begin{cases}2~~~~~~if ~~0\leq x\leq 1; 0\leq y\leq 1; 0\leq x +y\leq 1\\0~~~~~~elsewhere\end{cases}\]
find \(P (x\leq 3/4, y\leq 3/4)\)
Please help me find the limits ALGEBRAICALLY, I know how to find it out by geometric way. However, it works only for uniform function like this. I would like to know how to find it out by algebraic way so that I can work on any function.

Answer 1

After using geometry, I got
\[\int_0^{1/4}\int_0^{3/4}2dydx +\int_{1/4}^{3/4}\int_0^{1-x}2 dydx\]
How to work on it without the pic?

Answer 2

What do you mean by "without the pic" ?

Answer 3

Hence, when drawing the pic, we can find the upper and lower limits to put on the integral. However, it works if the function is uniform only.

Answer 4

if the function is conditional one, we HAVE TO find the limits algebraically; pic doesn't work.

Answer 5

I think solving the equation x+y<=1 should give us limits for each case

Answer 6

Show me, please.

Answer 7

Like for the first case, take y<=3/4 so 1-x<= 3/4 => x<=1/4
That gives limits for first term

Answer 8

Wait I got that all wrong

Answer 9

Now, suppose we don't have the pic and we don't know that the function can be break into 2 terms like that. How to do?

Answer 10

Normally beginning with something like this should give you the same expression you got. \[\int\limits_{0}^{3/4}\int\limits_{0}^{3/4} 2 dx dy [x+y \le 1] = \int\limits_{0}^{3/4}\int\limits_{0}^{\min(3/4;x)}2dx dy\]
I tried playing with it a little bit, but didn't get much...

Answer 11

:(

Answer 12

Are you saying that you want to figure out the bounds of a double integral w/o using a sketch of the region ?

Answer 13

Yes, Sir

Answer 14

that would be like searching for an object in the darkness w/o turning on the flashlight you're holding in your hand

Answer 15

Could you give me one valid reason for not using the sketch of the region of integration ?

Answer 16

`if the function is conditional one, we HAVE TO find the limits algebraically; pic doesn't work. `

why ?

Answer 17

My prof said that using picture as above works only for uniform function, for other functions, we have to find it out algebraically.

Answer 18

Let me post his way, it is so complicated so that I don't get it.

Answer 19

Okay..

Answer 20

Check this
http://math.stackexchange.com/questions/355383/algebraic-way-to-change-limits-of-integration-of-a-double-integral

Answer 21

Thanks for the link, I got it. :)

Answer 22

However, that is the method we use in cal3 with a fixed function, right? like our uniform function here.

Answer 23

I am still going through replies in that link...

Answer 24

That is 2 methods of integration. dy dx and dx dy. Both them rely on the pic.