Integrals

• Why use “$\int$” instead of “$\sum$” (Sigma)?
• “$\sum$” is used for discrete sums. (countable list of items, like $1+2+3+\dots +n)$.
• “$\int$” is used for continuous sums. (infinite number of infinitesimally small values over a continuous range).

Riemann Sum

$$ Area \approx \sum_{i=1}^nf(x_i^*)\cdot \Delta x $$

where $f(x_i^)$ is the height of the rectangle and $\Delta x$ is its width.*

Switching from “$\sum$” to “$\int$” involves taking a limit (continuous sum)

$$ \lim_{\Delta x \rightarrow 0} \sum_{i=1}^n f(x_i^*) \cdot \Delta x = \int_a^b f(x) dx $$

Fundamental Theorem of Calculus

• Slope: $\frac{dy}{dx}$
• Area under graph: $dy\times dx$
• $\div$ and $\times$ are opposites of each other
• So differentiating and integrating should be opposites of each other