In this section we discover that in trying to find the area under a curve or the distance traveled by a car, we end up with the same special type of limit.
We begin by attempting to solve the area problem: find the area of the region $S$ that lies under the curve $y = f(x)$ from $a$ to $b$. This means that $S$, is bounded by the graph of a continuous function $f$[where $f(x) ≥ 0$], the vertical lines $x = a$ and $x = b$, and the $x$-axis.
In trying to solve the area problem we have to ask ourselves: what is the meaning of the word area? This question is easy to answer for regions with straight sides. For a rectangle, the area is defined as the product of the length and the width. The area of a triangle is half the base times the height. The area of a polygon is found by dividing it into triangles and adding the areas of the triangles.
Recall that in defining a tangent we first approximated the slope of the tangent line by slopes of secant lines and then we took the limit of these approximations. We pursue a similar idea for areas. We first approximate the region $S$ by rectangles and then we take the limit of the sum of the areas of the approximating rectangles as we increase the number of rectangles. The following example illustrates the procedure.
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Use rectangles to estimate the area under the parabola $y = x^2$ from $0$ to $1$ (the parabolic region $S$).
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Let's apply the idea to the more general region $S$. We start by subdividing $S$ into $n$ strips $S_1, S_2, ..., S_n$ of equal width.
The width of the interval $[a, b]$ is $b − a$, so the width of each of the $n$ strips is
$$ Δx = \frac{(b − a)}{n} $$
These strips divide the interval $[a, b]$ into $n$ subintervals
$$ [x_0, x_1], [x_1, x_2], [x_2, x_3], ..., [x_(n-1), x_n] $$
where $x_0 = a$ and $x_n = b$. The right endpoints of the subintervals are
$$ x_1 = a + Δx,\\ x_2 = a + 2Δx,\\ x_3 = a + 3Δx,\\ .\\ .\\ . $$
and, in general, $x_i = a + iΔx$.
Now let's approximate the $i$th strip $S_i$ by a rectangle with width $Δx$ and height $f(x_i)$, which is the value of $f$ at the right endpoint. Then the area of the ith rectangle is $f(x_i) Δx$. What we think of intuitively as the area of S is approximated by the sum of the areas of these rectangles, which is
$$ R_n = f(x_1) Δx + f(x_2) Δx + ⋯ + f(x_n) Δx $$
