Integration MC Question - North Sydney Boys 2017 Trial (1 Viewer)

[fingerscrossed2019]

As the title says. Can someone help me with this question? The answer is C but I can't seem to understand why. My thinking is that the integral should be equal to 0, which rules out A and B. Then, looking at D I think it would calculate the area in some cases, but never calculate the integral. What am I missing?



Thanks in advance

 

[leehuan]

As correctly stated, A and B are good because they both equal to 0, soo they are ruled out.

C fails because by taking the odd function f(x) = -x^3, we have



D is good because using properties of definite integrals,



Hence C.

 

[fingerscrossed2019]

That's awesome and helps so much, thanks for your help!

 

[kawaiipotato]

You can also reason it like this:
If f takes non-negative values on [-a,0], then the first term is non-negative. The second term will be non-negative because of the absolute value.
Then the sum of those two is non-negative, meaning that if we had a function f with the previously mentioned properties, then the sum is non-negative. For a sum of non-negative terms to be zero, then both terms have to evaluate to zero individually ie. we claim that

which shouldn't be true for every odd function f when f is non-negative on [-a,0].