TRUE OR FALSE?

Question

if f(x) is an even function,
(above integral sign: 0  below integral sign: -a) ∫f(x)dx= (above integral sign: a  below integral sign: 0) -∫f(x)dx
 
true or false?

Answer 1

An even function is symmetrical to the y-axis.  Try a simple even function to see if the statement is true or not.  I would start with f(x) = x^2.  On the right hand side of the statement, you can take away the negative sign if you reverse your lower and upper limits.  If you take the integral from -a to 0 and compare it to taking the integral from a to 0 on an even function... you can tell from there.

 

In order to show prove algebraically that the statement is true, you need to do only two simple things.  First, on the left hand side, make the integral negative and reverse the upper and lower limits.  I'll show the result a few spaces down, so try to work it out yourself before looking:

 

 

 

 

 

 

 

 

 

 

(Upper: -a Lower: 0) -∫f(x)dx= (Upper: a Lower: 0) -∫f(x)dx

 

Now, since both sides of the equation are negative, they cancel out. (I used multiplication property of equality; multiplied both sides by -1 to cancel their negatives)

 

Your final result should be (Upper: -a Lower: 0) ∫f(x)dx= (Upper: a Lower: 0) ∫f(x)dx

 

This is one of the identities for integrals of even functions, so you are done!  The answer is true.

Answer 2

Difficult to write an answer by typing. I will list steps to find the answer

 

1. Switch limits from (-a,0) to (0,-a) .Switching limits puts a negative sign in front of the integral

2. Change variable y = -x, so dy = -dx, and f(y) = f(x) (cause f() is even). Limit changes to (0,a) for variable y and 

3. Negative sign in front of integral cancels the negative sign with dy. So we have a new integral 

4. limits are (0,a) of f(y)dy. 

 

If I read the choice of a negative sign in front of the integral, then the answer is false. I just worked this in my head so you should double check my logic on this.

Answer 3

It is false for even function

 

It is true for odd function