Two limited integration questions

[hhegab]

Peace!

I want to know the conditions that must be satisfied by a function
$$f(x)$$ for any of the following two cases to be true (each case independent from the other);

1- $$\int^a_{-a} f(x) dx = 2 \int^a_0 f(x) dx$$

2- $$\int^a_0 f(x) dx = \int^0_a f(x) dx$$

They gave me confusion when I was solving problems related to electric field and electric potential.

 

[quasar987]

I don't know if this condition is sufficient but a condition would be, if we write the integrals in terms of their primitives,

$$\int^a_{-a} f(x) dx = 2 \int^a_0 f(x) dx \Leftrightarrow \mathcal{F}(a) - \mathcal{F}(-a) = 2\mathcal{F}(a) \Leftrightarrow \mathcal{F}(-a) = -\mathcal{F}(a)$$

The condition is that it is true iff the primitive of f is a function F such that F(-a) = -F(a)

 

[marlon]

First case...

$$\int^a_{-a} f(x) dx = \int^0_{-a} f(x) dx + \int^a_0 f(x) dx$$

Then use the fact that :

$$\int^0_{-a} f(x) dx = - \int^{-a}_0 f(x) dx$$

and replace x by -x...the limit -a will then change to a because of this substitution. and dx will become -dx. Now f(x) becomes f(-x) and there are two possibilities. Either f(-x) = -f(x) or f(-x) = f(x)...you know what you will need to achieve so which one of the two is it...


Question 2 :

Just put the integral in right hand side to the left hand side and use the above property to get rid of the minus-sign...what do you get ?


regards
marlon

 

[hhegab]

Can you put like , first case is true if f(x) is even and if such and such...
I shall study your answer. But from my first reading I need more.
I need a condition to apply to f(x) so I can use each of the above properties.

hhegab