Content Basic Topics Calculus Function Differentiation Limit Range Main Topic Integration Quadrature Quadratic Equation Quadrature Formula Derivation

1. Topic – General Quadrature Formula
Subject Incharge: Dr. Dharm Raj Singh
Name: Mrinal Dev

2. Basic Topics
 Calculus
 Function
 Differentiation
 Limit
 Range
Main Topic
 Integration
 Quadrature
 Quadratic Equation
 Quadrature Formula
 Derivation

3. Calculus: Calculus, is the mathematical study of continuous
change, in the same way that geometry is the study of shape and
algebra is the study of generalizations of arithmetic operations.
Function: A function was originally the idealization of how a
varying quantity depends on another quantity.
Differentiation: Differentiation is a process of finding a function
that outputs the rate of change of one variable with respect to
another variable.

4. Limit: A limit is the value that a function (or sequence)
"approaches" as the input (or index) "approaches"
some value. Limits are essential
to calculus (and mathematical analysis in general) and
are used to define continuity, derivatives, and integrals.
It is of Two Types and they are:
1- Lower Limit:- The lower class limit of a class is the
smallest data value that can go into the class.
2- Upper Limit:- The upper class limit of a class is the
largest data value that can go into the class.
Range: Values can occur between the smallest and
largest values in a set of observed values or data points.
Given a set of values, or data points, the range is
determined by subtracting the smallest value from the
largest value.

6. Integration is the reverse of differentiation.
However:
If y = 2x + 3, dy/dx = 2
If y = 2x + 5, dy/dx = 2
If y = 2x, dy/dx = 2
So the integral of 2 can be 2x + 3, 2x + 5, 2x, etc.
For this reason, when we integrate, we have to add a
constant. So the integral of 2 is 2x + c, where c is a
constant.
A "S" shaped symbol is used to mean the integral of,
and dx is written at the end of the terms to be
integrated, meaning "with respect to x". This is the
same "dx" that appears in dy/dx .

7. To integrate a term, increase its power by 1 and
divide by this figure. In other words:
∫ xn dx = 1/n+1 (xn+1) + c
Examples:
∫ x5 dx = 1/6 (x6) + c

8.  In mathematics, quadrature is a historical term
which means determining area. Quadrature
problems served as one of the main sources of
problems in the development of calculus, and
introduce important topics
in mathematical analysis.

9. An equation where the highest exponent of the
variable (usually "x") is a square (2). So it will
have something like x2, but not x3 etc.
A Quadratic Equation is usually written
ax2 + bx + c = 0.
Example: 2x2 + 5x − 3 = 0.
Quadrature Equation is also known as Newton’s
Forward Interpolation Formula.

10. The Gauss–Kronrod quadrature formula is an adaptive method for
numerical integration. ... It is an example of what is called a
nested quadrature rule: for the same set of function evaluation points, it has
two quadrature rules, one higher order and one lower order (the latter called
an embedded rule).
The Formula is given by:
y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎
𝒖(𝒖−𝟏)
𝟐!
𝟐 𝒚 𝟎 +
𝒖(𝒖−𝟏)(𝒖−𝟐)
𝟑!
𝟑 𝒚 𝟎 + ⋯

11. Let I = ydx where y = f (x)
Also assume that f (x) be given for certain equidistant values of x, say
x0, x1, x2, x3,….xn. Let the range (b-a) be divided into n equal parts,
each of width h, so that
h =
𝑏 − 𝑎
𝑛
Thus, we have
x0 = a, x1 = a + h, x2 = a + 2h, … xn = a + nh = b
Now , let
yk = f (xk), k = 0,1,2…n
Consider, I = y dx = ydx
b
a

b
a

0
0
x nh
x



12. We have
y = ∫ 𝒇 𝒙 = 𝒚 𝟎 + 𝒖 𝒚 𝟎 +
𝒖(𝒖−𝟏)
𝟐!
𝟐 𝒚 𝟎 +
𝒖(𝒖−𝟏)(𝒖−𝟐)
𝟑!
𝟑 𝒚 𝟎 + ⋯
where u =
du =
dx = hdu
𝑥 − 𝑥0
ℎ
1
ℎ



13. Approximating y by Newton’s forward formula taking limit of
integration becomes 0 to n.
I = h y0 + u y0+
𝑢(𝑢−1)
2!
2 𝑦0 +
𝑢(𝑢−1)(𝑢−2)
3!
3 𝑦0 + ⋯ du
n
= h y0u +
𝒖 𝟐
2
y0 +
𝒖 𝟑
𝟑
−
𝒖 𝟐
𝟐
2!
2 y0 +
𝑢4
4
−𝑢3
+𝑢2
3!
3y0 + …
0
=h ny0 +
𝒏 𝟐
2
y0 +
𝒏 𝟑
𝟑
−
𝒏 𝟐
𝟐
2!
2y0 +
𝑛4
4
−𝑛3
+𝑛2
3!
3y0 + …
0
n


14. I = nh y0 +
𝒏
2
y0 +
𝒏(𝟐𝒏−𝟑)
12
2y0 +
𝒏 𝒏−𝟐 𝟐
24
3y0+…
This is called General Quadrature formula.