# |∫f(x)dx|(from a to b=∫|f(x)|(from a to b) then , a (a) exactly one root in (a, b)(b) at least one root in (a, b)(c) no root in (a, b)(d) None of the abovecan someone please help with i

Sai Ram Charan
31 Points
3 years ago
Sorry for answering this question very lately! I’m seeing this question just now!
I think you know that $\int_{a}^{b}f(x)dx$ is positive if and only if the function $y=f(x)$ lies completely above X-axis in the interval$(a,b)$ . Similarly, that integral will be negative only if it lies completely below X-axis in $(a,b)$.
So, if you have a function like $y=\sin (x)$, you can see that it lies above X-axis in$(0,\pi)$ and it lies below X-axis in $(\pi,2\pi)$. But, the function $y=\left | \sin (x) \right |$ is above X-axis in $(0,\pi)$ and also in $(\pi,2\pi)$. So, now you’ll understand:
$\int_{0}^{2\pi}\left | \sin (x)\right |dx =4 \textup{ but} \left | \int_{0}^{2\pi}\sin (x)dx \right |=0$ so, here the integrals are not equal
So,f(x)= sin(x) has one root in $(0,\pi)$
So, we can conclude that if the integrals are equal, f(x) has no root in (a,b)