Question

For a function $f\left(x\right)\ge 0\forall xϵ(a,b);a<b$,
$\left|{\int }_a^{b}f\right(x\left)dx\right|\le {\int }_a^b|f\left(x\right)|dx$.

• A. True
• B. False

Answer 1

The correct option is A True

Well, Let’s understand what this question is saying by the following figure made by Chintu.

Let’s say this is the graph of f(x).
We know that ${\int }_a^{b}f\left(x\right)dx$ gives us the algebraic sum of the area.
${\int }_a^{b}f\left(x\right)dx=$ 1st Blue region + Red region + 2nd Blue region
Let 1st Blue region be = $B_1$
Red region be = R
2nd Blue region be = $B_2$
So,${\int }_a^{b}f\left(x\right)dx=B_1+R+B_2$
We know as the red region is below the X -axis, the area thus calculated from this will be negative
where as $B_1\&B_2$ are positive.
$|{\int }_a^{b}f\left(x\right)dx|$ will be the difference of R & $B_1+B_2$.
$|{\int }_a^{b}f\left(x\right)dx|$ is nothing but the absolute value of this difference.
Now, have a look at another graph y = |f(x)|

In this graph ${\int }_a^{b}f\left(x\right)|dx$ gives the area under the curve.
${\int }_a^b|f\left(x\right)|dx=B_1+R+B_2$
Since here, R is also above the X- axis along with $B_1\&B_2$. The area will be the sum and not difference of all the regions.
And we can clearly say that the difference of some numbers will always be less than the sum of these numbers.
So,$\left|{\int }_a^{b}f\right(x\left)dx\right|\le {\int }_a^b|f\left(x\right)|dx$.
Now the question comes where will the equality sign hold good. Can you guess?
Yes, it’ll happen when the graph is always on either side of X- axis, i.e either above X- axis throughout or
below X- axis . For example for $f\left(x\right)=e^x$