Question

What are the symmetry properties of finding areas using integrals?

Answer 1

If `f` is an even function (symmetric about the y-axis), then

`int_{-a}^a f(x) dx=2int_0^a f(x) dx`.

If `f` is an odd function (symmetric about the origin), then

`int_{-a}^a f(x) dx=0`.

Symmetries can be used to simplify computation of definite integrals. Let us look at the following examples.

Example 1 (Even Function)

#int_{-1}^1(3x^2+1) dx =2int_0^1(3x^2+1) dx=2[x^3+x]_0^1=2(2-0)=4#

Example 2 (Odd Function)

`int_{-pi/3}^{pi/3}{sin theta}/{sqrt{cos^2 theta+1}} d theta=0`

I hope that this was helpful.