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.
The Riemann sum approaches the signed area. In your picture, the green area is positive, and the red area is negative. The Riemann sum should approach $0$, which is the accurate signed area for $f(x)=-x$ on the interval $[-1,1]$. If you don't like that, try $f(x)=|x|$.