Question

I'm a little confused about how my calculus textbook is discussing integrating symmetric functions in two different ways: (Uploading images from my book in a minute.)

Answer 1

I know one is dealing with symmetric functions across the y-axis, but why does one method of finding the area take the absolute value of both portions of the integral which has been divided in half across the y-axis, (which makes sense to me) and the other says that the total area of the odd functions is just zero? Shouldn't it be treated in the same way, i.e. the absolute value of both sides?

Answer 2

because areas do not subtract.
look at the second picture. you are given an example wherein
1) for the graph you need to find the integration
2) for the same graph, the area under the curve.

integration is a tool to find the area under the graph but while doing so, it may "subtract" the negative areas. which we do not do in compouting the areas

Answer 3

Yeah, that makes sense. I got lulled into "Area under the curve" in terms of entirely real values (Like how you can't have a negative length), not negative areas, and assumed that the general nature of the integral would treat it like summing the absolute values of said areas. So for general reference, the definite integral of some thing that has a portion of "negative" space is not equivalent to the area in practical terms? (That itself is iffy, I know that that will change with real-world applications and stuff, but if just treating it like "the area of this graph" with no physical allegory, I think it should hold)

Answer 4

yep