#18 Practice book

[ducduyaktt]

let f be a function that the graph of f is semicircle with endpoints (a,0) and (b,0) where a<b ( a, b are real numbers)

18. The improper integral :[ integral(from a to b) f(x)f'(x)dx ] is necessarily ZERO.

Could anyone please explain it for me? Thanks


And ONE more question :

which letter is not homomorphic to letter C? J, N, S, O, U

Thanks so much

[D13E12]

Hi answer for Q no. 18 is by integrating the integrand f(x)f'(x)dx we get (f^2)(x)/2| with limits to substitute as a to b. so, the integral equal to ((f(b))^2-(f(a))^2)/2=0, because f(a)=0, f(b)=0.
note:-It's not an improper integral. integral is called improper only when one or both of the limits are infinity.
I do not the know answer to the second question. But if you come to know please do post it (with explanation).
Thanks.
cheers,
Siva Nagi