Riemann Integral

The Riemann integral is the Integral normally encountered in Calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue Integral), but are unlikely to be encountered outside the confines of advanced mathematics texts.

The Riemann integral is based on the Jordan Measure, and defined by taking a limit of a Riemann Sum,

$$\int_b^a f(x)\,dx \equiv \lim_{\max\Delta x_k\to 0} \sum_{k=1}^n f(x_k^*)\Delta x_k$$ (1)

$$\int\!\!\!\int f(x,y)\,dA \equiv \lim_{\max\Delta A_k\to 0} \sum_{k=1}^n f(x_k^*,y_k^*)\Delta A_k$$ (2)

$$\int\!\!\!\int\!\!\!\int f(x,y,z)\,dV \equiv \lim_{\max\Delta V_k\to 0} \sum_{k=1}^n f(x_k^*,y_k^*,z_k^*) \Delta V_k,$$ (3)

where $a\leq x\leq b$ and $x_k^*$, $y_k^*$, and $z_k^*$ are arbitrary points in the intervals $\Delta x_k$, $\Delta y_k$, and $\Delta z_k$, respectively. The value $\max\Delta x_k$ is called the Mesh Size of a partition of the interval $[a,b]$ into subintervals $\Delta x_k$.

As an example of the application of the Riemann integral definition, find the Area under the curve $y=x^r$ from 0 to $a$. Divide $(a,b)$ into $n$ segments, so $\Delta x_k = {b-a\over n} \equiv h$, then

$$f(x_1) = f(0) = 0$$ (4)

$$f(x_2) = f(\Delta x_k) = h^r$$ (5)

$$f(x_3) = f(2\Delta x_k) = (2h)^r.$$ (6)

By induction

$$f(x_k) = f([k-1]\Delta x_k) = [(k-1)h]^r = h^r (k-1)^r,$$ (7)

so

$$f(x_k)\Delta x_k = h^{r+1}(k-1)^r$$ (8)

$$\sum_{k=1}^n f(x_k)\Delta x_k = h^{r+1} \sum_{k=1}^n (k-1)^r.$$ (9)

For example, take $r = 2$.

$\sum_{k=1}^n f(x_k)\Delta x_k = h^3 \sum_{k=1}^n (k-1)^2$
$\quad = h^3 \left({\,\sum_{k=1}^n k^2 - 2\sum_{k=1}^n k + \sum_{k=1}^n 1}\right)$
$\quad = h^3 \left[{{n(n+1)(2n+1)\over 6} - 2 {n(n+1)\over 2} + n}\right],$	(10)

so

$$I \equiv \lim_{n\to \infty} \sum_{k=1}^n f({x_k}^*)\Delta x_k = \lim_{n\to \infty} \sum_{k=1}^n f(x_k)\Delta x_k$$

$$= \lim_{n\to \infty} h^3 \left[{{n(n+1)(2n+1)\over 6} - 2 {n(n+1)\over 2} + n}\right]$$

$$= a^3 \lim_{n\to \infty} \left[{{n(n+1)(2n+1)\over 6n^3} - {n(n+1)\over n^3} + {n\over n^3}}\right]$$

$$= {\textstyle{1\over 3}} a^3.$$ (11)

See also Integral, Riemann Sum

References

Kestelman, H. ``Riemann Integration.'' Ch. 2 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 33-66, 1960.