Midpoint Rule Calculator

How to Use the Midpoint Rule Calculator

The Midpoint Rule Calculator is a tool designed to help you approximate definite integrals of functions over an interval using the midpoint rule. Follow these step-by-step instructions to effectively use the calculator for your mathematical computations.

Step 1: Enter the Lower Limit

In the input section of the calculator, locate the field labeled ‘Lower Limit (a)’. This is where you should input the beginning value of the integration range. Ensure your input is a number within the specified range of -1,000,000 to 1,000,000 and use up to four decimal places if necessary. This field is required, so it must not be left empty.

Step 2: Enter the Upper Limit

Next, find the input field labeled ‘Upper Limit (b)’. Here, you need to input the end value of the integration range. Similar to the lower limit, the value must be between -1,000,000 and 1,000,000 and can have up to four decimal places. This field is also required for the computation to proceed.

Step 3: Specify the Number of Subintervals

Identify the field called ‘Number of Subintervals (n)’. Enter the number of subintervals you wish to use for the approximation. The number must be at least 1 and can go up to 10,000. This value determines the precision of your approximation, as more subintervals typically result in a more accurate result. Ensure this field is filled as it is mandatory.

Step 4: Select the Function

Select a function from the dropdown menu labeled ‘Select Function f(x)’. The available options include x², x³, sin(x), cos(x), eˣ, and 1/x. This selection is crucial as it defines the function over which the integration will be performed. Ensure a function is chosen because it is a required input.

Step 5: Calculate the Results

Once all inputs are entered correctly, the calculator will compute the results for you. Here are the fields you can expect to see:

• Δx (Width of each subinterval): This value is calculated using the formula: (upperLimit – lowerLimit) / subintervals. It shows the width of each subinterval in your approximation.
• Approximate Integral: The calculator provides an approximate value of the integral using the formula: deltaX * sum(map(range(0, subintervals), i => evaluate(function, {x: lowerLimit + (i + 0.5) * deltaX})).
• Estimated Error: The potential error in the approximation is calculated as: abs((deltaX^2 * (upperLimit – lowerLimit)) / 24) giving you an understanding of the precision of the result.

Each value is displayed with a precision of up to six decimal places to provide detailed mathematical insights.

Conclusion

The Midpoint Rule Calculator is a powerful tool for approximating the integral of selected functions over a defined interval. By following these steps, you ensure accurate inputs and achieve meaningful results, aiding in your mathematical exploration and problem-solving.