Integral Calculus

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.