Riemann Sums

• draw in rectangles and find the sum of their areas
• options: left, right, and midpoint
• over/under estimate
  • inscribed rectangles (inside curve) → underestimate
  • circumscribed rectangles (outside curve) → overestimate

Trapezoids

• not technically Riemann sums but the same idea
• draw trapezoids to find the sum of areas
• $A=w\frac{a+b}{2}$

Over/Under Estimates

Riemann Left/Right depends on first derivative

Direction	Left	Right
Increasing	Under	Over
Decreasing	Over	Under

Trapezoid and Riemann midpoint depends on concavity

Concavity	Trap	Midpoint
Down	Under	Over
Up	Over	Under

Limit

$$\begin{gathered} {\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f(x_i)\Delta x \\ \Delta x=\frac{b-a}{n} \\ {x}_i=a+\Delta x\ast i \end{gathered}$$