Trapezoidal rule 5

-
Find the value of 
By (i) Trapezoidal rule and (ii) Simpsons 1/3rd rule  taking n=6.

Solution:-
h= (π/2 -0) / 6 = π/12
Y= f(x) = √cos x. 

X              Y
0               1
π/12         0.96593
π/6           0.86603
π/4           0.70711
π/3           0.5
5π/12       0.25881
π/2            0

(I) using Trapezoidal rule
Integration of y dx = h/2 [ (y0 + yn) + 2 (y1 + y2 +........ Y(n-1)]
 => π/24 [ (1+0) + 2(0.96593+ 0.86603 + 0.70711 + 0.5 + 0.25881 ) ]
 => π/24 [ 1 + 2× 3.29788]
=> π/24 [ 1 +  6.59576]
=> π/24 × 7.59576
=> 0.99428

(ii) Simpsons 1/3rd rule. 
Integration of y dx = h/3[ (y0 + yn) + 4 (y1 + y3 +........ Y(n-1) +(y2+ y4 +........ Y(n-2) ]

=> π/36 [(1+0)+4(0.96593+0.70711 +0.25881) + 2 (0.86603+  0.5 )]
=> π/36 [1 + 4× 1.93185 + 2× 1.36603 ]
=> π/36 [1 + 7.7274 + 2.73206]
=> π/36 × 11.45946
=> 1.00003


Comments

Post a Comment

Popular posts from this blog

planar and non planar

-
Image
What is the difference between planar and non- planar graph?  PLANNER GRAPH :- • A graph in which the direction of current or direction or of Edge is given is called planar graph.  • planar graph is also known as oriented graph.  Non-Planar graph:- •  A graph in which the direction of current age is not given is called Non - Planar graph.  • Non- planar graph is also known as unoriented graph. 
Read more

ammeter and voltmeter

-
Ammeter and Voltmeter are connected in series and parallel respectively. Why? Ans. In series connection, the same current flows through all the components.  Ammeter aims at measuring the current flowing through the circuit and hence, it is  connected in series, so that the same current that is there in the circuit flows through  it and gets measured. Voltmeter aims at measuring the amount of potential difference across a resistor or  series of resistor. A voltmeter measures the potential difference of the circuit and it  has high internal resistance. When the voltmeter is connected in parallel with a circuit  component, the amount of current passing through the voltmeter is very less.  Therefore, the current through the circuit is unaltered.
Read more

zener diode and working principle

-
Image
What do you mean by zener diode, explain its working principle with neat and clean diagram?  Zener diode is a special type of diode designed to reliably allow current to flow "backward" (that is Reverse Biased) when a certain set reverse voltage known as a zener voltage is reached. zener diode are manufactured with a great variety of Zener voltage and some even variable.  Working principle:- When a PN junction diode is highly doped ,  then concentration of impurity atoms will be high in the Crystal. The higher concentration of impurity atoms cause the highest concentration of ions in the depletion layer. Hence for some applied reverse Bias Voltage the width of depletion layer , becomes than that  in a normally doped diode due to thinner depletion layer, voltage gradient for electric field strength across the depletion layer is quite high. if the reverse voltage is continued to increase, after a certain applied voltage , the electron from the covalent bonds  with...
Read more