What are the symmetry properties of finding areas using integrals?

Question

Christopher Allers · Accepted Answer

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.

Emilia Aguilar · Answer

undefined The symmetry properties of finding areas using integrals are:

1. **Symmetry about the x-axis:** If a function $ f(x) $ is even (symmetric about the y-axis), then the area between the curve and the x-axis from $ x = -a $ to $ x = a $ is twice the area from $ x = 0 $ to $ x = a $.

2. **Symmetry about the y-axis:** If a function $ f(x) $ is odd (symmetric about the origin), the area between the curve and the x-axis from $ x = -a $ to $ x = a $ is zero.

3. **Symmetry about the origin:** If a function $ f(x) $ is symmetric about the origin, then the area between the curve and the x-axis from $ x = -a $ to $ x = a $ is twice the area from $ x = 0 $ to $ x = a $.