▶️Integral Calculus
Integral calculus is a process to calculate the AREA
between a function and the X-axis (or Y-axis).
Core idea of Integral Calculus
Refer to Khan academy: Introduction to integral calculus
Riemann Sums
Riemann Sums
A Riemann sum is an approximation of the area under a curve by dividing it into multiple simple shapes (like rectangles or trapezoids).
Riemann Sums Notation
Riemann Sums Notation
Refer to Khan academy: Definite integral as the limit of a Riemann sum
The letter ʃ
(reads as "esh" or just "integral") is called the Integral symbol/sign
.
Calculate Riemann Sums
Calculate Riemann Sums
Finding indices m & n
: It's meant to find the i
for Σ
sums:
For
Left Sums
orMidpoint Sums
:i
starts from0
ends withsubdivisions - 1
For
Right Sums
:i
starts from1
ends withsubdivisions
Finding xi
: With equally spaced points (left/right/mid), the xi
is a Geometric series
of those points, which the rate is the 𝚫x
. We're gonna find the right pattern/equation for xi
, so that we can plug xi
into f(x)
.
Finding f(xi)
: Just to plug in the Geometric series expression of xi
into f(x)
, and make it as a function in terms of i.
Left & Right Riemann Sums Approximation
Refer to Maths is fun: Integral Approximations
Left Riemann Sum
: take the Left boundary value of Δx to be the rectangle's height.Right Riemann Sum
: take the Right boundary value of Δx to be the rectangle's height.
As you can see, they would be either Over-estimated or Under-estimated. Neither of these approximations would be called a good one, normally.
Midpoint Sums Approximation
It's an enhancement to the Left sums and Right sums, it takes the midpoint value, and sometimes makes better approximation.
Example
Example
It's easy to find the
Δx=2
.Then let's find the
f(x𝖎)
. It's actually a progress to find theArithmetic Sequence
.So the sequnce is
S(𝖎) = a + 𝖎·Δx = 2 + 2𝖎
, wherea
represents the firstx
value which is2
.So
x𝖎 = S(𝖎) = 2+2𝖎
Takes it back to the function and gets:
f(x𝖎) = |2+2i-5| = |2i -3|
Example
Example
How to calculate Riemann Sums
How to calculate Riemann Sums
Refer to Khan academy: Rewriting definite integral as limit of Riemann sum
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