# Math Help - Calculus Proof

1. ## Calculus Proof

Prove if f is continuous and non-negative in [a,b], then

the integral from a to b of f(x)dx >= 0.

Any thoughts...thanks.

2. Originally Posted by jzellt
Prove if f is continuous and non-negative in [a,b], then

the integral from a to b of f(x)dx >= 0.

Any thoughts...thanks.
Surely if $f$ is continuous and nonnegative in $[a, b]$, then its graph lies on or above the $x$ axis in this region.

What does this tell you about the area enclosed by the graph and the $x$ axis in this region?

3. Yeah I see that, and to answer your question, that tells me that the area is positive.

But, I'm not sure how to give a rigourous proof of this...

4. If it's obvious, it doesn't need proving.

5. It does if it's going to be a potential question on my exam...

6. What I am saying is, if you give an obvious argument that convinces the reader, as mine has, then you have completed the proof.

Proofs don't necessarily have to be rigorous.

7. Or, to really impress your teacher:

Imagine dividing the interval from a to b into n intervals and construct the Riemann sum for this function. Since each "rectangle" has has positive length base (by the construction of the Riemann sum that is always true) and positive length height (because $f(x)\ge 0$ and the height is always f(x*) for some x*), the area of each rectangle is greater than or equal to 0 and so is the Riemann sum. Since every Riemann sum is greater than or equal to 0, the integral, the limit of those Riemann sums, is greater than or equal to 0.

8. Originally Posted by Prove It
If it's obvious, it doesn't need proving.
I have moved this thread to the Analysis subforum where, despite my signature, even the obvious requires rigorous proof.

@OP: I'm not sure what theory of integration you have studied. I suggest considering the Riemann integral in the first instance (although you might want to consider the Riemann–Stieltjes integral or even the Lebesgue integral depending on what level you're studying at).