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Simpson’s Rule Calculator + Online Solver With Free Steps
The online Simpson’s Rule Calculator is a tool that solves the definite integrals in your calculus problems using the Simpson’s Rule. The calculator takes the information regarding the integral function as input.
Definite integrals are the closed integrals in which endpoints of intervals are defined. The calculator provides the numerical value, symbolic form, error graph, and method comparisons for the given definite integral.
What Is a Simpson’s Rule Calculator?
A Simpson’s Rule Calculator is an online tool specifically designed to evaluate the definite integrals via Simpson’s rule.
Solving integrals always remains a challenging task because it is a time-consuming and tiring process. Additionally, to avoid having inaccurate results, one must have a good base in integration-related concepts.
The most common technique to evaluate the definite integral is solving the integral and then putting the limit values. But there is another easier technique that does not use any kind of integration known as Simpson’s rule.
Simpson’s rule is a method in which we divide the interval into further sub-intervals and define a width between each sub-interval. It uses the function values to evaluate the definite integral.
This handy calculator uses the same method to determine the values of definite integrals. It is one of the best available tools as it is relatively faster and delivers error-free results.
How To Use the Simpson’s Rule Calculator?
You can use the Simpson’s Rule Calculator by putting the details of definite integrals in their respective boxes. After this, a detailed solution will be presented in front of you with just a single click.
Follow the detailed instructions given below while using the calculator.
Step 1
Put the function that needs to be integrated into the first box located on the right side with the label “interval.”
Step 2
Then enter the lower and upper limits of integration in the tabs From and To, respectively.
Step 3
The last step is to click the Evaluate button to get the final result of the problem.
Output
The output of Simpson’s Rule Calculator has multiple sections. The first section is the input interpretation where the user can cross-check that the input is correctly inserted.
Then the result section displays the numerical value obtained after solving the integral. Also, it provides you with the symbolic form of Simpson’s rule. Then it plots the Error vs Interval graph. There are two different graphs because there are two types of errors.
An absolute error means the difference between the calculated and actual value while a relative is a percentage error obtained by dividing the absolute error by the actual value. Lastly, it provides a detailed comparison of both errors obtained using Simpson’s rule with errors in all other methods.
How Does the Simpson’s Rule Calculator Work?
This calculator works by finding the approximate value of the given definite integral over a specific interval. This interval is further divided into n subintervals of equal width.
This calculator along with the value of the integral also calculates the relative error bound across each interval. The working of this calculator can be acknowledged by understanding the concept behind Simpson’s Rule.
What Is Simpson’s Rule?
Simpson’s rule is the formula that is used to approximate the area under the curve of a function f(x) that results in finding the value of the definite integral. The area under the curve using the Riemann sum is calculated by dividing the area under the curve into rectangles. However, the area under the curve is divided into parabolas using Simpson’s rule.
The definite integral is calculated by using integration techniques and by applying the limits but sometimes these techniques can not be used to evaluate the integral or there is not any particular function that is to be integrated.
Therefore, Simpson’s rule is used to approximate the definite integrals in these scenarios. This rule is also known as Simpson’s third rule, which is written as Simpson’s ⅓ rule.
Simpson’s Rule Formula
Simpson’s rule is the numerical method that gives the most accurate approximation of an integral. If there is a function f(x)=y over the interval [a,b] then the Simpson’s rule formula is given by:
\[ \int_{a}^{b} f(x) \,dx \approx (h/3)[f(x_{0})+4 f(x_{1})+2 f(x_{2})+…+2 f(x_{n-2})+4 f(x_{n-1})+f(x_{n})]\]
Where x0=a and xn=b, n is the number of subintervals in which the interval [a,b] is divided and h=[(b-a)/n] is the width of the subinterval.
The idea behind this rule is to find the area using quadratic polynomials. The parabolic curves are used to find the area between two points. It is contrary to the trapezoidal rule which uses straight line segments to find the area.
Simpson’s third rule is also used to approximate the polynomials. This can be utilized up to third-order polynomials.
Simpson’s Rule Error Bound
Simpson’s rule does not give the exact value of the integral. It provides the approximate value, hence an error is always there which is the difference between the actual value and approximate value.
The error value is given by the following formula:
\[Error bound= \frac{M(b-a)^5}{180n^4}\]
Where $|f^{(4)}(x)| \le M$.
How To Apply Simpson’s Rule
The approximate value of the integral $\int_{a}^{b} f(x) \,dx$ can be found using Simpson’s rule by first recognizing the values of the limits a and b of the given interval and the number of subintervals, which is given by the value of n.
Then determine the width of each subinterval by using the formula h=(b-a)/n. The width of all subintervals must be equal.
Afterward, the interval [a,b] is divided into n subintervals. These subintervals are $[x_{0},x_{1}], [x_{1},x_{2}], [x_{2},x_{3}],…., [x_{n-2},x_{n-1}], [x_{n-1},x_{n}]$. The interval must be divided into even numbers of subintervals.
The required value of the integral is obtained by plugging all the above values into Simpson’s rule formula and simplifying it.
Solved Examples
Let’s look at some problems resolved using Simpson’s Calculator for a better understanding.
Example 1
Consider the below-given function:
\[ f(x) = x^{3} \]
Integrate it over the interval x=2 to x=8 with the interval width equal to 2.
Solution
The solution to the problem is in several steps.
Exact Value
The numerical value is:
2496
Symbolic Form
The symbolic form of Simpson’s rule for the problem is:
\[ \int_{2}^{10} x^{3} dx \approx \frac{1}{3} \left( 8 + 2 \sum_{n=1}^{4-1} 8(1 + n)^{3} + 4 \sum_{n=1}^{4} 8(1 + 2n)^{3} + 1000 \right) \]
\[ \int_{x_{1}}^{x_{2}} f(x) dx \approx \frac{1}{3} h \left( f(x_{1}) +2 \sum_{n=1}^{4-1} f( 2hn + x_{1} ) + 4 \sum_{n=1}^{4} f(h(-1+2n) + x_{1}) + f(x_{2}) \right) \]
Where $f(x)=x^{3}$, $x_{1}=2$, $x_{2}=10$ and $h=(x_{2}-x_{1})/(2\times4) = (10-2)/8 =1$.
Method Comparisons
Here is some comparison between different methods.
Method | Result | Absolute error | Relative error |
Midpoint | 2448 | 48 | 0.0192308 |
Trapezoidal rule | 2592 | 96 | 0.0384615 |
Simpson’s rule | 2496 | 0 | 0 |
Example 2
Find the area under the curve from x0 to x=2 by integrating the following function:
f(x) = Sin(x)
Consider the interval width equal to 1.
Solution
The solution to this problem is in multiple steps.
Exact Value
The numerical value after solving the integral is given as:
1.41665
Symbolic Form
The symbolic form of Simpson’s rule for this problem is as follows:
\[ \int_{2}^{10} sin(x)dx \approx \frac{1}{6} \left( 8 + 2 \sum_{n=1}^{2-1} sin(n)+ 4 \sum_{n=1}^{2} sin(\frac{1}{2} (-1 + 2n) ) + sin(2) \right) \]
\[ \int_{x_{1}}^{x_{2}} f(x) dx \approx \frac{1}{3} h \left( f(x_{1}) + 2 \sum_{n=1}^{2-1} f( 2hn + x_{1} ) + 4 \sum_{n=1}^{2} f(h(-1+2n) + x_{1}) + f(x_{2}) \right) \]
Where f(x)=sin(x), x1=0, x2=2 and $h=(x_{2}-x_{1})/(2\times2) = (2-0)/4 =\frac{1}{2}$.
Method Comparisons
Method | Result | Absolute error | Relative error |
Midpoint | 1.4769 | 0.0607 | 0.0429 |
Trapezoidal rule | 1.2961 | 0.1200 | 0.0847 |
Simpson’s rule | 1.4166 | 0.005 | 0.0003 |