What are the symmetry properties of finding areas using integrals?

Question

What are the symmetry properties of finding areas using integrals?

1 Answer

Impact of this question

8581 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-28T14:07:50

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

$\int_{- a}^{a} f (x) d x = 2 \int_{0}^{a} f (x) d x$ $int_{-a}^a f(x) dx=2int_0^a f(x) dx$ .

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

$\int_{- a}^{a} f (x) d x = 0$ $int_{-a}^a f(x) dx=0$ .

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

Example 1 (Even Function)

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

Example 2 (Odd Function)

$\int_{- \frac{π}{3}}^{\frac{π}{3}} \frac{sin θ}{\sqrt{{cos}^{2} θ + 1}} d θ = 0$ $int_{-pi/3}^{pi/3}{sin theta}/{sqrt{cos^2 theta+1}} d theta=0$

I hope that this was helpful.

Science

Math

Humanities

... and beyond

What are the symmetry properties of finding areas using integrals?

1 Answer

Related questions

Impact of this question