📝 Introduction to Loops with Integrals#

Utility of loops#

Branching causes specific blocks of code to be executed only under certain conditions. The conditions are articulated as logical expressions.

How do you write loops?#

The simplest version of loop is a “for loop”.

for looping_variable in sequence:
    code_block

More complex structures are possible with “while statements”.

while logical_expression:
    # code_block is repeated until logical expression is false
    code_block

Just like if-statements, for-loops can be nested.

EXAMPLE: Let x be a two-dimensional array, [[10, 20],[30 40]]. Use a nested for-loop to sum all the elements in x.

import numpy as np

x = np.array([[10, 20], [30, 40]])
print(x)

n, m = x.shape
s = 0
for i in range(n):
    for j in range(m):
        s += x[i, j]

print(s)

[[10 20]
 [30 40]]
100

Using loops in integration#

Many engineering applications require integration, which can be understood as calculating the area under a curve. For example, consider the following:

  • estimating the force of water against a bridge or dam

  • determining the amount of time a reactor should operate for a given chemical to be produced

  • processing audio or electrical signals into meaningful input

  • determining the force required to move an object a given distance

For example, to deposit materials on graphene, one could observe the progress of experiments over time to find the amount of mass deposited to optimize the conditions for deposition.

The Riemann integral is the simplest method of approximating the area under a curve. One simply

  • divides the area under the curve into rectangles of equal width \(h\) and height \(f(x_i)\),

  • calculates the area for each rectangle using \(A_i = hf(x_i)\), and

  • sums the areas of each rectangle between some smaller value of \(x\) called \(a\) and a larger value of \(x\) called \(b\).

Mathematically, this process is expressed as

\[\int_{x_i}^{x_{i+1}} f(x) dx= hf(x_i) + O(h^2).\]