Riemann Sums

We can use Riemann Sums to approximate the area under a curve.

For some function $f(x)$ over $[a,b]$:

• We divide the distance between $a$ and $b$ into $n$ slices
  • The width of each slice $\Delta x$ is given by the formula $\Delta x=\frac{b-a}{n}$
• The right sum approximates using the right value of each slice
  • $A_{\text{right}}=\sum _{i=1}^{n}f(a+i\Delta x)\Delta x$
• The left sum approximates using the left value of each slice
  • $A_{\text{left}}=\sum _{i=0}^{n-1}f(a+i\Delta x)\Delta x$
• The midpoint sum approximates using the middle value of each slice
  • $A_{\text{mid}}=\sum _{i=1}^{n}f(a+(i-\frac{1}{2})\Delta x)\Delta x$

To get a more accurate approximation, we can use:

• The trapezoid rule, which is the mean of the left and right sums
  • $A_{\text{trapezoid}}=\frac{1}{2}(A_{\text{left}}+A_{\text{right}})=\frac{\Delta x}{2}[f(x_0)+2f(x_1)+2f(x_2)+...+2f(x_{n-1})+f(x_n)]$
• Simpson’s rule
  • $A_{\text{simpson}}=\frac{\Delta x}{3}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+2f(x_4)+...+4f(x_{n-1})+f(x_n)]$
    • This approximately simplifies to $\frac{\Delta x}{3}(A_{\text{trapezoid}}+2A_{\text{midpoint}})$

The Definite Integral

As we increase the number of slices $n$, the resolution of our area becomes increasingly accurate. Taking the limit of the sum as $n$ approaches infinity, we get the definite integral.

$${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(a+i\Delta x)\Delta x$$

The input $a+i\Delta x$ is often expressed as $\Delta x_i^{\ast }$.

For example, we can find the area under $f(x)=\sin (e^x)$ from $0$ to $\frac{\pi }{2}$ using the limit of a Riemann sum.

$${\int }_0^{\pi /2}\sin (e^x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^n\sin (e^{0+i(\frac{\pi /2}{n})})(\frac{\pi /2}{n})$$