How a Definite Integral can have the same value as just of one of the points of the integration interval?

Question

I was evaluating the improper integral

$$I=\int_1^\infty \frac{1}{x^2}\,dx$$

and found $I = 1$.

But, I am confused now: the integral is an operation that sums all the values of the integrand function $f(x) = 1/x^2$ in the whole interval $x\in [1,\infty[$ .

If one calculates $f(1)= 1/1^2 = 1$, it turns out that this is the same value for the whole integral, but evaluated at only one point of the integration interval!

The function is $f(x) > 0$ in the whole integration range, so no cancelations between positive and negative values occur.

In short (no formal language): how a sum can be equal to just one term of the whole sum being that all terms are positive?

Answer 1

It's a coincidence. You're basically asking, "how can the area of a shape happen to be equal to its height at one end?"

Answer 2

The integral is not the sum of all the numbers $1/x^2$ for $x\ge1$. That sum is infinite. This is easier to see if you look at the integral over some finite interval, say $[1,2]$. The integral is the limit of a Riemann sum, and in no way is it equal to the sum of $1/x^2$ for al $1\le x\le 2.$ When you add up an uncountable number of positive numbers, you always get $\infty.$