Introduction to Simpson's Rule Calculator:
Simpson's rule calculator is an online tool that helps you to evaluate the approx value of a given definite integral function. It is used to find the area under a curve from the given complex quadratic polynomial function on an interval.
When you solve the Simpson rule question by hand it takes time to get the solution or you may get confused during calculation. That’s why we introduce simpson rule calculator that keeps you away from all these difficulties and provides you the solution with just one click.
What is Simpson's Rule?
Simpson rule is a numerical integration process of calculating the approximate value of the area under a graph over a bonded region. The Simpson rule is the extension of the trapezoidal rule because it finds the complex integral problem of the second order(quadratic polynomial).
The Simpson rule has two formulas that is used to give the approximate value for different type of quadratic polynomial functions under a parabola.
What is Simpson's 1/3 Rule?
For a function f(x) that divides into n subinterval over an interval [a,b], the Simpson's ⅓ rule formula used by the Simpson's rule calculator is given as:
$$ \int_a^b f(x) dx \approx \frac{\Delta x}{3} (f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + … + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)) $$
- h=b-a/n is the width of the subinterval
- f(x) is the function and x0,x1,x2,x3… is the value of subinterval of a given function
- [a,b] is the interval of a given function
What is Simpson's 3/8 Rule?
Simpson's 3/8 Rule is a method for approximating the definite integral of a function for cubic polynomials because it is particularly useful when the subintervals are a multiple of 3.
It gives a more accurate result than Simpson's 1/3 rule. It is used for odd number values only. The simpsons rule calculator uses ⅜ rule also if the question is supposed to solve by this formula,
$$ \int_a^{b} f(x) dx \;=\; \frac{3 \Delta x}{8} [f(x_0) + 2(f(x_3) + f(x_6) + f(x_{n-1})) + f(x_n)] $$
How to Calculate Simpson's Rule?
Simpson rule can be calculated with the help of two formulas which are the Simpson 1/3 rule Simpson 3/8 rule for complex integral functions for quadratic polynomial functions or cubic polynomial functions respectively.
Let's see examples of the Simpson rule where we take both the rules and evaluate them so that you know how the simpson's 1/3 rule calculator solves such questions easily.
Example of Simpson’s Rule:
An example of simpson’s rule is given below so that you may know how the simpson’s rule calculator solves such problems easily.
Example: Find the following integral using simpson’s rule with n = 4.
$$ \int_0^4 (1 + x^2) dx $$
Solution:
Step 1:
Determine the values from the given integral function. In this example, we have given
$$ n \;=\; 4,\; a \;=\; 0\; and\; b \;=\; 4,\; f(x) \;=\; 1 + x^2 $$
Step 2:
Find the width of subinterval h as h=b-a/n
$$ h \;=\; \frac{4-0}{4} \;=\; 1 $$
$$ h \;=\; 1 $$
Step 3:
Then find the subinterval of a given function f(x) such as:
$$ f(x) \;=\; f(x_0), f(x_1), f(x_2), f(x_3), f(x_4) $$
$$ x_0, x_1 \;=\; 1, x_2 \;=\; 2, x_3 \;=\; 3, x_4 \;=\; 4 $$
$$ f(0) \;=\; 1 + 0^2 \;=\; 1 $$
$$ f(1) \;=\; 1 + 1^2 \;=\; 2 $$
$$ f(2) \;=\; 1 + 2^2 \;=\; 5 $$
$$ f(3) \;=\; 1 + 3^2 \;=\; 10 $$
$$ f(4) \;=\; 1 + 4^2 \;=\; 17 $$
Step 4:
Apply the 1/3 Simpson rule and modify it as per the given function and its subinterval such as,
$$ \int_a^b f(x) dx \;=\; \frac{h}{3} [(y_0 + y_n) + 4(y_1 + y_3 + y_5 + … + y_{n-1}) + 2(y_2 + y_4 + y_6 + … + y_{n-2})] $$
Step 5:
Add all the values in the 1/3 Simpson rule formula to get the solution,
$$ \int_0^4 (1 + x^2) dx \approx \frac{1}{3} [1 + 4(2 + 10) + 2(5) + 17] $$
Step 6:
Therefore the solution of a given definite integral over a subinterval n=4.
$$ \int_0^4 (1 + x^2) dx \approx \frac{1}{3} [1 + 48 + 10 + 17] $$
$$ \int_0^4 (1 + x^2) dx \approx \frac{1}{3} \times 76 \;=\; \frac{76}{3} \approx 25.333 $$
Example: Find the following integral using Simpson’s ⅜ rule with n = 3.
$$ \int_0^3 e^x dx $$
Solution:
Step 1:
Determine the values from the given integral function. In this example, we have given
$$ n \;=\; 3,\; a \;=\; 0\; and\; b \;=\; 3,\; f(x) \;=\; e^x $$
Step 2:
Find the width of subinterval h as:
$$ h \;=\; \frac{b-a}{n} $$
$$ h \;=\; \frac{3-0}{3} \;=\; 1 $$
$$ h \;=\; 1 $$
Step 3:
Then find the subinterval of a given function f(x) such as:
$$ f(x) \;=\; f(x_0),\; f(x_1),\; f(x_2),\; f(x_3) $$
$$ x_0 \;=\; 0,\; x_1 \;=\; 1,\; x_2 \;=\; 2,\; x_3 \;=\; 3 $$
$$ f(0) \;=\; e^0 \;=\; 1 $$
$$ f(1) \;=\; e^1 \;=\; e $$
$$ f(2) \;=\; e^2 \;=\; e^2 $$
$$ f(3) \;=\; e^3 \;=\; e^3 $$
Step 4:
Apply the ⅜ Simpson rule and modified it as per the given function and its subinterval such as,
$$ \int_0^3 e^x dx \approx \frac{3h}{8} [f(0) + 3(f(1) + f(2)) + f(3)] $$
Step 5:
Add all the values in the 3/8 Simpson rule formula.
$$ \int_0^3 e^x dx \approx \frac{3.1}{8} [1 + 3(e + e^2) + e^3 ] $$
Step 6:
Therefore the solution of given definite integral over a subinterval n=3.
$$ \int_0^3 e^x\; dx \approx \frac{3}{8} [1 + 3e + 3e^2 + e^3] $$
How to Use Simpson's Rule Calculator?
The simpson rule calculator has a simple design that makes it easy for you to know how to use it for the evaluation of complex integral functions. You need to follow our guidelines before using it to avoid inconvenience. These guidelines are:
- Enter the complex integral function of the subinterval in the given input field.
- Enter the value of the upper and lower limit in the input field for the Simpson rule.
- Enter the subintervalue in the required field.
- Recheck the given complex integral expression before clicking the calculate button to start the evaluation process in simpsons rule calculator.
- Click the “Calculate” button to get the result of your given complex integral over a single interval.
- If you are trying our simpson's 1/3 rule calculator for the first time then you can use the load example to learn more about this concept.
- Click on the “Recalculate” button to get a new page for finding more examples with solutions to complex integral problems using the Simpson rule.
Final Result of Simpson Rule Calculator:
The simpson calculator gives you the solution from a given complex integral function when you add the input into it. It contains as:
- Result Option
When you click on the result option the simpson 1/3 rule calculator gives you a solution to the given integral function
- Possible Steps
When you click on it, this option will provide you with a solution where all the calculations of the Simpson's rule procedure are given in steps
Advantages of Simpsons Rule Calculator:
The simpson's rule error calculator provides you with many advantages as it helps you to calculate complex integral problems and gives you solutions without any trouble. These advantages are:
- The simpson's method calculator is a free tool so you can use it free to find complex integral problem solutions without paying anything
- It is a manageable tool that can manage various types of integral problems that have quadratic polynomial or cubic functions because it has a built-in Simpson rule to solve these types of problems easily.
- Our simpson calculator gives you conceptual clarity for the integral process when you use it for practice to solve more examples.
- It saves the time that you consume on the calculation of complex integral problems manually and provides solutions in a few second
- Simpson 1/3 rule calculator is a reliable tool that provides you with accurate solutions whenever you use it to calculate complex integral problems without any errors in calculation.
- Simpson's rule Calculator enables you to use it multiple times for the evaluation of complex integral problems over a bounded region.
