1. ## Integral Help

So I don't really understand integrals as much as I had hoped... can someone help me figure out this really simple problem?

Find the lower and upper integrals of $\int_0^1 f(x) dx$ if f : [0,1] $\rightarrow$ R is given by

f(x):=
x if x $\in$ Q $\cap$ [0,1]
-x if x $\in$ (R \ Q) $\cap$ [0,1]

2. Originally Posted by thaopanda
So I don't really understand integrals as much as I had hoped... can someone help me figure out this really simple problem?

Find the lower and upper integrals of $\int_0^1 f(x) dx$ if f : [0,1] $\rightarrow$ R is given by

f(x):=
x if x $\in$ Q $\cap$ [0,1]
-x if x $\in$ (R \ Q) $\cap$ [0,1]
Inside every interval, no matter how small, there exist both rational and irrational numbers. If, while doing your "Riemann sums", you evaluated f(x) only at irrational numbers you function would look exactly like y= x. What is the integral of that from 0 to 1? If, while doing your "Riemann sums", you evaluated f(x) only at rational numbers you function would look exactly like y= -x. What is the integral of that from 0 to 1?

3. for y = x, is it $\frac{1}{2}$?

for y = -x, is it $\frac{-1}{2}$?

so would that mean they're the upper and lower sums?