Please help me with integrals!

Question

If f(x) and g(x) are continuous on [a, b], which one of the following would be false?

Options:

1. 
2. 
3. 
4. 

Answer 1

Hi Veranna C.,

So, you are being asked to carefully scan these statements for identities.

Option 1 is an identity of a type, since the integral of ANYTHING over a zero-length interval is zero. There can be no area to integrate! (Maybe someone could quibble, what if the argument is infinite, so what is 0 times infinity? Undefined!)

Option 2 is an identity, it's just the distribution of addition across an integration operation. Since each "bit" added in the integral is the sum of the 2 individual "bits" from f and g, the sum of the integrals is the integral of the sums, so to speak.

Option 4 is an identity (similar to Option 2), the distribution of subtraction across integration.

But Option 3: AHA! should say "minus one times the integral", not "one minus the integral". So that's the false one.

A note for the future: Whenever you have a question of this format (find the false statement, etc.) you need to shift mentally into "low gear": the falsification is likely to be subtly hidden in language, or in improper domain of functions (i.e. impossible function calls), or something concealed in negative vs. positive numbers, or false quadratic roots, etc.. When in doubt, throw a few "extreme" values of x (or whatever variable) in to test (i.e. those values outside known function sign reversal points, etc. -- so, for |x-5|, you would try 0 and 10, perhaps ) including test values form intervals between any subset of such points -- so, for expressions with both |x+5| and |x-5| in, you would try -10, 0, and 10, perhaps.

-- Cheers, -- Mr. d.