This document provides an overview of integral calculus and numerical integration methods. It defines integral calculus as the branch of calculus dealing with functions to be integrated. Integration is described as the reverse process of differentiation where the integral calculates the area under a curve. Numerical integration methods like the trapezoidal rule, Simpson's 1st rule, and Simpson's 2nd rule are explained along with their applications, accuracy, and requirements. Definite and indefinite integrals are also defined. In summary, the document outlines key concepts in integral calculus and compares common numerical integration techniques.

1. SHRI RAMSWAROOP MEMORIAL UNIVERSITY
SESSION – 2023-2024
TEACHER ASSESSMENT
MATHEMATICS (BMA4012)
NAME – VAIBHAV SAHU
ROLL NO. – 202110101110079
COURSE- BTECH CS-43
SUBMITTED TO- DR. VIMLESH
SUBMITTED BY- VAIBHV SAHU

2. INTEGRAL CALCULUS :
It is the branch of calculus which deals
with functions to be integrated.
INTEGRATION :
 Integration is the reverse process of
differentiation.
 The function to be integrated is referred to
as integrand while the result of an integration
is called integral.
 The integral is equivalent to the area under
the curve.

3. The integral symbol is an elongated S –
denoting sum, was introduced by Leibniz, who
named integral calculus as calculus
summatorious.
Numerical integration is carried by the
numerical methods and they are of three types:
Trapezoidal rule
Simpson’s 1 st rule
Simpson’s 2 nd rule

4. DEFINITE INTEGRAL : defined by the limit
values a & b of the independent variable.
INDEFINITE/PRIMITIVE INTEGRAL :
An integral with no restrictions imposed
on its independent variable.

5. ⦁ It is applicable for equal intervals.
⦁ The error is of order h2.
⦁ The accuracy can be improved by increasing
the no. of intervals & by decreasing the value
of h.
⦁ In this rule, y(x) is a linear function of x.
⦁ In general, trapezoidal rule is less accurate
when compared with Simpson's rule.

6. ⦁ It is also known as Simpson's one-third (1/3)
rule.
⦁ It is applicable for even intervals.
⦁ The error is of order h4.
⦁ In this rule, y(x) is a polynomial of degree 2.
⦁ It uses 3 data points.

7. ⦁ It is also known as Simpson's 3/8 th rule.
⦁ The error is of order h5.
⦁ In this rule, y(x) is a polynomial of degree 3.
⦁ It is applicable for the intervals which is
multiple of 3.
⦁ It uses four data points.

8. TRAPEZOIDAL RULE:
SIMPSON’S 1 st RULE :
∫
𝑥0
𝑛
𝑓( 𝑥) 𝑑𝑥 = ℎ
3 0 n 2 4
[( y + y ) + 2( y + y +…) +
4(y1 3
+ y +…) ]
SIMPSON’S 2 nd RULE :
𝑥𝑛
∫ 𝑓 (𝑥) 𝑑𝑥 =
3ℎ
8
[ 𝑦0 + 𝑦𝑛 + 3 𝑦1 + 𝑦2 + 𝑦4 + 𝑦5 + ⋯
𝑥0
+ 2 𝑦3 + 𝑦6 + 𝑦9 + ⋯ ]
∫𝑥0 2 0
𝑥𝑛
𝑓 𝑥 𝑑𝑥 = ℎ
[ 𝑦 + 𝑦𝑛 + 2(𝑦 + 𝑦 + ⋯ + 𝑦𝑛
1 2 − 1)]

9. Where…
x0 = initial value of x,
y0 = initial value of y,
xn = final value of x,
yn = final value of y,
h = interval distance,
h = 𝑏−𝑎
𝑛
n = no. of intervals.

10. It helps to
 Find the area.
 Locate the centroid.
 Find the arc length of a graph.
 Find the surface area of a solid.
 Find the volume of a solid figure.
 Solve for the work done.
 Solve the moment of inertia.