Jingle belll

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.