Introduction to Midpoint Rule Calculator:
Midpoint rule calculator is an online tool that helps you to evaluate the approx value of a given area over a closed interval using midpoint rule. It is used to compute the estimated value of each subinterval of a given area in a graph.
Our midpoint rule online calculator is a useful online source to find the definite integral of each subinterval. It especially solve the definite integral problems which cannot be solved easily by analytic calculus methods.
What is the Midpoint Rule?
Midpoint rule is a numeral analysis method in which you can calculate the estimated value of an area over a bounded region when you divide it with each of its subintervals. It is known as midpoint rule or midpoint approximate.
It is a type of riemann sum integral method in which you find the area of a rectangle from the given interval.
Formula of Midpoint Rule:
The formula of midpoint rule consists of a continuous function f(x) over a closed interval [a, b] that divides each subinterval.
$$ \int_a^b f(x) dx \approx \Delta x \sum_{i=1}^{n} f(x_i^*) $$
- a and b: the lower or the upper limits
- Δx: change in each subinterval
- f(x): the continuous function of definite integral
- n: the number of subinterval its values vary as n = 0,1,2,....
How to Calculate Midpoint Rule For Integrals?
To calculate the midpoint rule for integrals in steps, you shoudl use the midpoint rule formula. Here is a step-by-step guide about how to solve the given midpoint question.
Step 1: Identify the function f(x) that you want to integrate and the interval [a, b] over which you want to integrate.
Step 2: Choose the number of subintervals n that divide the interval [a, b].
Step 3: Calculate the width of each subinterval Δx,
$$ \Delta x \;=\; \frac{b-a}{n} $$
Step 4: For each subinterval i, the midpoint is calculated as:
$$ x_i^* \;=\; x_0 + x_{\frac{1}{2}}, x_1 + x_{\frac{2}{2}}, x_2 + x_{\frac{3}{2}},... $$
$$ For\; i \;=\; 1, 2,...,n $$
Step 5: Put the above values in the midpoint rule formula and simplify it to get a solution.
$$ M_n \;=\; \sum_{i=1}^{n} f(m_i) \Delta x $$
Step 6:Evaluate the given integral and apply its upper and lower limits in solution.
Step 7: The final approximation of the integral is given by,
$$ Error\; in\; M_n ≤ \frac{M(b-a)^3}{24n^2} $$
Alternatively, You can use our midpoint rule calculator, which is highly efficient at finding the definite integral for each subinterval. It provides accurate results within seconds.
Practical Example of Midpoint Rule:
An example of the midpoint rule with a solution is given below to understand how the midpoint rule integral calculator works.
Example: Use Midpoint Rule to Estimate the Following Using Four Subintervals
$$ \int_0^1 x^2 dx $$
Solution:
Identify the given function f(x) and limits,
$$ f(x) \;=\; x^2,\; a \;=\; 0,\; b \;=\; 1 $$
$$ \int_0^1 x^2 dx $$
Find the value of each subinterval,
$$ Δx \;=\; \frac{b-a}{n} $$
$$ \Delta x \;=\; \frac{1-0}{4} \;=\; \frac{1}{4} $$
As the limit is between 0 and 1 so the subinterval is between (0,1),
$$ [0, \frac{1}{4}],\; [\frac{1}{4}, \frac{1}{2}], [\frac{1}{2}, \frac{3}{4}],\; and\; [\frac{3}{4}, 1] $$
For f(x*) value we use this rule f(x0 + x1/2) for each subinterval,
$$ \int_a^b f(x) \;=\; \Delta x \left(f \left( \frac{x_0 + x_1}{2} \right) + f \left(\frac{x_1 + x_2}{2} \right) + f \left(\frac{x_2 + x_3}{2} \right) + … + f \left(\frac{x_{n-2} + x_{n-1}}{2} \right) + f \left(\frac{x_{n-1} + x_{n}}{2} \right) \right) $$
The midpoints of these subintervals are,
$$ \left( \frac{1}{8},\; \frac{3}{8},\; \frac{5}{8},\; \frac{7}{8} \right) $$
The formula of midpoint rule,
$$ \int_a^b f(x) dx \approx \Delta x \sum_{i=1}^{n} f(x_i^*) $$
Put the value of Δx and f(x*) in the above formula,
$$ M_4 \;=\; \frac{1}{4} . f (\frac{1}{8} ) + \frac{1}{4} . f (\frac{3}{8}) + \frac{1}{4} . f (\frac{5}{8}) + \frac{1}{4} . f(\frac{7}{8}) $$
Simplify it,
$$ \frac{1}{4} . \frac{1}{64} + \frac{1}{4} . \frac{9}{64} + \frac{1}{4} . \frac{25}{64} + \frac{1}{4} . \frac{21}{64} \;=\; \frac{21}{64} $$
After evaluating the limit of the given function we get,
$$ \int_0^1 x^2 dx \;=\; \frac{1}{3} $$
For the error approximation value,
$$ \biggr| \frac{1}{3} - \frac{21}{64} \biggr| \;=\; \frac{1}{192} \approx 0.0052 $$
How to Use the Midpoint Rule Approximation Calculator?
Midpoint rule integration calculator has a simple design that makes it easy for you for evaluating the midpoint rule problems. Follow some simple steps that are given as:
- Enter the given definite integral function in the given input field (that you want to evaluate to find its approx value).
- Add the upper and the lower limit of the midpoint integral in the next input field.
- Add the number of subinterval around which you evaluate the approx value.
- Check the given integral function value before clicking on the calculate button of midpoint rule formula calculator to start the evaluation process.
- Click the “Calculate” button to get the result of your given midpoint rule problem.
- If you are trying our midpoint rule error calculator for the first time then you can use the load example to learn more about this concept.
- The “Recalculate” button brings back to the home page where you can find more solutions of midpoint rule problems.
Output from Midpoint Rule Online Calculator:
The midpoint rule calculator gives you the solution from a given function when you give it an input. It may included as:
- Result Option:
When you click on the result option, it gives you a solution of definite integral function for each interval.
- Possible Steps:
When you click on it, this option will provide you step by step solution of the estimated value of definite integral.
- Plot Option:
It will give you a graphical representation of a definite integral function solution.
Benefits of Using Midpoint Rule Integral Calculator:
Midpoint rule approximation calculator provides you with tons of benefits as it helps you to calculate the estimated value of each subinterval of a given function and give you solutions without any trouble. These benefits are:
- It is a free-of-cost tool so you can use it for free to find midpoint problem with solutions without spending.
- Midpoint rule integration calculator gives you conceptual clarity for the midpoint rule process when you use it for practicing more examples.
- It saves the time and effort that you consume on the calculation of complex definite integral functions for finding the estimated value manually.
- Midpoint rule formula calculator is an adaptable tool that provides you with accurate solutions whenever you use it to calculate integral for each subinterval without any mistakes in the evaluation process.
- Our tool enables you to use it multiple times for the evaluation of midpoint rule problems.
- Midpoint rule error calculator is a handy tool because you can access it through an online platform from anywhere with the help of the internet only.
