# Integral Calculus

##### May 31, 2020

- tags
- Math

These are the two ways we commonly think about definite integrals: they describe an accumulation of a quantity, so the entire definite integral gives us the net change in that quantity.

^{1}

## Why Integral Calculus #

Figure 1 represents 2 graphs of `y = cos(x)`

. Let’s say we would
like to calculate the area of
\( x\_1 \)
. We could calculate
the area by aproximation, for example, Graph B is filled with the area we would
like to calculate, so we could divide this area by equal sections of
\(\Delta x_n\) from `a`

to `b`

rectangles, then we could calculate the area of
these rectangles by \(f(x_i) * \Delta x_n\) where \(f\) is the area of each of
the rectangles. We do this for each rectangle then sum them up: \(\sum_{i=1}^n f(x_i) *
\Delta x_n\). This will give us an approximation of our area, we could have a
better approximation by having our \(\Delta x_n\) smaller, but this implies that
our `n`

becomes bigger and bigger. The smaller \(\Delta x_n\) gets, the more `n`

approaches infinity.

We could use \(\liminf\) of `n`

as `n`

approaches ∞ or \(\Delta x_n\) as it
gets very small.

```
set multiplot layout 1, 2 title "f(x) = -x ** 2 + 4"
set terminal pngcairo enhanced color size 350,262 font "Verdana,10" persist
set linetype 1 lc rgb '#A3001E'
set style fill transparent solid 0.35 noborder
f(x) = -x ** 2 + 4
set title "A"
plot f(x) with lines linestyle 1
set title "B"
set style fill transparent solid 0.50 noborder
plot f(x) fs solid 0.3 lc rgb '#A3001E'
unset multiplot
```

The idea of getting better and better approximations is the what constitutes Integral Calculus.

: Source: Exploring accumulation of change ↩︎