Question

Question from Riemann integral. Need help.

Answer 1

the riemann integral is the sum of the areas under the curve, therefore, if the functions has negative values, we will also get negative areas, that in the total sum in the interval from a to b, this can be positive, negative or zero, and it's not necessary that the functions f is 0 for every x in [a, b]

Answer 2

for example, \(f(x) = sin(x)\) in the interval \([0, 2\pi]\), we have \[\int_{0}^{2\pi} sin(x) dx = 0\]

Answer 3

we have integral over a to b, |f(x)|dx=0 in [a,b]

Answer 4

@M4thM1nd

Answer 5

Oh i didn't see the | |. From the definition of modulus:
|f(x)| = f(x), if f(x) > 0
|f(x)| = -f(x), if f(x) < 0

Answer 6

@ganeshie8

Answer 7

@Alchemista

Answer 8

so... \[\int_{a}^{b} \left| f(x) \right|dx = -\int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx\], where f(x) is negative in [a, c] and f(x) is positive in [c,b]

Answer 9

that means that the only way this integral be 0 is if f(x) = 0 in [a, b]

Answer 10

we know that \[|\int\limits_{a}^{b}f dx| \le \int\limits_{a}^{b}|f|dx\]

Answer 11

@M4thM1nd I don't get the reasoning properly, kindly explain.

Answer 12

both the answers are no..

Answer 13

however thanks, you tried to help.@M4thM1nd