Surprising result?

Dr.Peterson

Elite Member
It looks to me like a matter of symmetry; the curve under which you are integrating is symmetrical about the point (pi/4, 1/2).

The area in each case is half of a rectangle with base pi/2 and height 1, because the curve bisects that rectangle. And that happens because f(pi/2 - x) = 1 - f(x).

• topsquark

Dr.Peterson

Elite Member
Yes, certain definite integrals are much easier than their indefinite integrals ...

I saw the answer purely visually; the last graph was most obvious, then I saw the symmetry in the others, and then I realized how to see that in the function itself. It's all about cofunctions.

apple2357

Full Member
I can see your transformations explanation works, i can't quite see why f(pi/2-x) is the same as f(x) but reflected in the way it is ( as below)
If it was f(x-pi/2) i would understand that as a simple translation but is f(pi/2-x) is a combination of transformations? Is this a translation and a reflection? How did you come up with f(pi/2-x) = 1-f(x) as the explanation for the graphical observation?

Dr.Peterson

Elite Member
I'd call it a double reflection, first around x=pi/4 and then around y=1/2. Equivalently, and the way I described it initially, it is actually a rotation around the point (pi/4, 1/2). In the same way, reflecting in both y=0 and x=0 amounts to rotating by 180 degrees about the origin.

Replacing x with pi/2 - x reflects in x=pi/4, and replacing y with 1-y (that is, subtracting the result from 1) reflects in y=1/2. These are worth pondering until you see why!

apple2357

Full Member
I'd call it a double reflection, first around x=pi/4 and then around y=1/2. Equivalently, and the way I described it initially, it is actually a rotation around the point (pi/4, 1/2). In the same way, reflecting in both y=0 and x=0 amounts to rotating by 180 degrees about the origin.

Replacing x with pi/2 - x reflects in x=pi/4, and replacing y with 1-y (that is, subtracting the result from 1) reflects in y=1/2. These are worth pondering until you see why!
So If we reflect y=f(x) in x=a, we get the curve y=f(−x+2a) ?

Is this because if you want to move every point to a position the same distance the other side of x=a we end up at x+2(a-x). Which gives us 2a -x
Is that how you would explain it?

Dr.Peterson

Elite Member
Yes, that is one good way to explain this fact.

And, of course, the same fact is why the other reflection is 1 - y.