Integration as a Limit and Bounds for Area

Split $[a,b]$ into $n$ strips of width $h=(b-a)/n$. A right-endpoint rectangle sum is $R_n=h\sum_{k=1}^{n}f(a+kh)$. For a continuous function, its limit as $n\to\infty$ is $\int_a^b f(x)\,dx$.

A left-endpoint sum instead uses $a,a+h,\ldots,b-h$. The sample points differ, but as the width tends to zero both sums approach the same integral. The limiting process removes the approximation error.

For a nonnegative increasing function, the left-endpoint height is the minimum on each strip and the right-endpoint height is the maximum. Thus the left sum is a lower bound and the right sum an upper bound. For a decreasing function, reverse their roles.

These bounds require information about the function between the sampled points. Increasing values in a table do not by themselves prove the function is increasing throughout every strip.

For an increasing function and equal widths, the two sums have all interior terms in common. Subtracting gives $R_n-L_n=h[f(b)-f(a)]$. This is the width of a valid bracket for the area.

Doubling the number of equal strips halves $h$ and therefore halves this upper-minus-lower gap. It does not make an individual approximation exact, but it makes the bracket tighter.

For a function that changes direction, split at turning points or use actual minima and maxima within each strip. Endpoints alone may not bound the heights.

For the same equally spaced ordinates, the trapezium estimate is $(L_n+R_n)/2$. It is the midpoint of the rectangle bracket, but the direction of its own error depends on curvature, not whether the graph is increasing.

For a curve that is concave up, a chord lies above the curve, so trapezia overestimate the signed integral on that interval. For concave down, they underestimate. If curvature changes, inspect separate intervals rather than assuming one rule applies everywhere.

Below the x-axis, negative heights contribute negative signed areas to the integral. To find total geometric area across axis crossings, split at the zeros and take the positive magnitude of each region. A limit of signed rectangle sums still gives signed accumulation.