Question

when f(x) is even is the function always 2?

Answer 1

If f(x) is even, what is \[\int\limits_{-3}^{3}f(x)dx?\] . what is the average value of f(x)on the interval x=-3 to x=3?

Answer 2

this is the part b) to a question. Part a) told us that \[\int\limits_{0}^{3}f(x)dx=6\]

Answer 3

If f(x) is an even function, it means that f(x)=f(-x).
This means that by symmetry:
\(\int_{-b}^b f(x)dx = 2\int_0^b f(x)dx\)

Answer 4

how did you get that, im kind of lost

Answer 5

While we're on the subject, an odd function has the property:
\(f(-x)=-f(x)\)
Example: \(sin(x), tan(x), x^3,x^5\), are all odd, while
\(cos(x), sec(x), x^2,x^4,x^6... \)are all even.
and hence by symmetry:
\int_{-b}^b f(x)dx = 0

Answer 6

or is that just a formula?

Answer 7

It's by symmetry.

As you can see in the drawing, area A = area B,
so Area A+B = 2*Area B