5.2 - Evolution of Integration

Riemann sums

• A way to approximate (and eventually obtain) an integral
  • Using techniques like LRAM, MRAM, RRAM, and the Trapezoidal Rule, we can estimate integrals
  • As we create more and more partitions, we can estimate integrals even better
  • Eventually, we will be drawing infinitesimally small integrals and getting an exact area
• Taking a Riemann sum involves drawing $n$ rectangles under a curve
  • Eventually, we take the limit of these rectangles as the number of them ($n$) approaches infinity and find their combined area.

Riemann sum formula and integral notation

• Notice that the $\mathrm{\Delta }x$ is the length of each equal partition.
• Some problems might involve going from this to integral notation: ${\displaystyle {\int }_a^{b}f(x)dx}$. Substitute $c_k$ for $x$ to convert between forms.

Some facts

• All continuous functions are integrable (able to be integrated).