Concept preview

What you will be able to do

1. 01Approximate the area under a curve (a definite integral) with left, right, and midpoint Riemann sums using $n$ rectangles of equal width $\Delta x = \dfrac{b-a}{n}$.
2. 02Determine whether a left or right Riemann sum over- or under-estimates the true area, based on whether the function is increasing or decreasing on the interval.
3. 03Explain that as the number of rectangles $n\to\infty$ (so $\Delta x\to 0$) every Riemann sum converges to the exact area — the definite integral $\displaystyle\int_a^b f\,dx$.

We want the area between a curve $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$. Unless that region is a familiar shape, we cannot read its area off a formula — so we approximate it with rectangles. First slice the interval $[a,b]$ into $n$ equal pieces. Each piece has width $\Delta x = \dfrac{b-a}{n}$; this common width is the base of every rectangle. Over each slice we stand up a rectangle whose height is the value of $f$ at some chosen sample point inside that slice. Summing the rectangle areas — each is (height) $\times \Delta x$ — gives an estimate of the region’s area. That estimate is a Riemann sum.