I'm having trouble with theoretical understanding of the Riemann sum with this explanation/definition from Thomas' Calculus. I checked Wikipedia and it seems to state virtually the same.:
On each subinterval we form the product $f(c_k)*∆x_k$. This product is positive, negative, or zero, depending on the sign of $f(c_k)$. When $f(c_k) > 0$, the product $f(c_k)*∆x_k$ is the area of a rectangle with height $f(c_k)$ and width $∆x_k$. When $f(c_k) < 0$, the product $f(c_k)*∆x_k$ is a negative number, the negative of the area of a rectangle of width $∆x_k$ that drops from the x axis to the negative number $ƒ(c_k)$.
Finally we sum all these products to get:
$$ S_p = \sum_{k=1}^{n}{f(c_k)}∆x_k $$
… Any Riemann sum associated with a partition of a closed interval [a, b] defines rectangles that approximate the region between the graph of a continuous function ƒ and the x-axis. Partitions with norm approaching zero lead to collections of rectangles that approximate this region with increasing accuracy
To illustrate the problem, suppose we want to approximate the area between $f(x) = -x$ and the x axis on the interval [-1; 1]. The area is 1, but the Riemann sum should give something close to 0:
Is the statement that any Riemann sum with the norm approaching 0 approximates the area with increasing accuracy correct? It seems not, since in the example above the area tends to 0 as the norm approaches 0, which is not "increasing accuracy". Does it miss the part that one should take the absolute values of the rectangles' areas?
Thank you.