Midpoint Rule Calculator

Midpoint Rule Calculator

Enter a definite integral, and the calculator will approximate its value using the midpoint (mid-ordinate) rule, providing step-by-step calculations.

Enter a Function:

W.R.T:

Lower Limit:

Upper Limit:

Number of rectangles:

An online midpoint rule calculator allows you to estimate a definite integral using the midpoint rule. This calculator also provides an approximation of the area under a curve compared to left, right, or trapezoidal sums.

What is the Midpoint Rule?

In mathematics, the midpoint rule approximates the area between the graph of a function f(x) and the x-axis by summing the areas of rectangles whose heights are determined at the midpoint of each subinterval.

Mid Point Rule Calculator

You can also use an online Riemann Sum Calculator to estimate definite integrals with midpoints, trapezoids, and left/right endpoints.

Midpoint Rule Formula

For a general n subintervals, the midpoint rule is:

$$ \int_a^b f(x) dx \approx \Delta x \left[ f\left(\frac{x_0 + x_1}{2}\right) + f\left(\frac{x_1 + x_2}{2}\right) + \dots + f\left(\frac{x_{n-1} + x_n}{2}\right) \right] $$

Where Δx = (b - a)/n is the width of each subinterval.

Example

Approximate the integral $ \int_1^4 \sqrt{x^2 + 4} \, dx $ using n = 5 subintervals.

Solution:

Δx = (4 - 1)/5 = 0.6. Divide [1, 4] into 5 subintervals: 1, 1.6, 2.2, 2.8, 3.4, 4.

Calculate midpoints and evaluate f(x) at each:

f((1+1.6)/2) = f(1.3) = √(1.3² + 4) = 2.3853
f((1.6+2.2)/2) = f(1.9) = √(1.9² + 4) = 2.7586
f((2.2+2.8)/2) = f(2.5) = √(2.5² + 4) = 3.2015
f((2.8+3.4)/2) = f(3.1) = √(3.1² + 4) = 3.6891
f((3.4+4)/2) = f(3.7) = √(3.7² + 4) = 4.2059

Sum the function values and multiply by Δx:

0.6 × (2.3853 + 2.7586 + 3.2015 + 3.6891 + 4.2059) ≈ 9.7444

This gives the approximate area under the curve. An online integral calculator can also compute this directly.

Midpoint Rule Error Bound Formula

The error in the midpoint approximation is bounded by:

$$ |E_M| \le \frac{K (b-a)^3}{24 n^2} $$

Where:

E_M = error in the midpoint rule
n = number of subintervals
f''(x) ≤ K on [a, b]

How the Midpoint Rule Calculator Works

Input:

Enter a function f(x) with upper and lower limits [a, b]
Specify the number of rectangles/subintervals
Click "Calculate"

Output:

Calculates Δx and midpoint values
Displays step-by-step evaluation of the function at midpoints
Provides the approximate area under the curve

FAQs

Is the midpoint rule more accurate than the trapezoidal rule?

The midpoint rule is generally more precise than the trapezoidal rule for smooth functions, according to composite error bounds. However, specific cases may vary.

Why is the trapezoidal rule less accurate?

It uses linear approximations, whereas the Simpson rule uses quadratic approximations. The trapezoidal method can underestimate or overestimate the area depending on the function curvature.

How do you determine the midpoint in a Riemann sum?

For each subinterval, take the midpoint x = (x_i + x_{i+1})/2, then evaluate f(x) at these midpoints to compute rectangle heights.

How do you compute error bounds?

Find the maximum of |f''(x)| on [a, b], then use the formula |E_M| ≤ K(b-a)^3 / (24 n^2).

Conclusion

Use an online midpoint rule calculator to approximate integrals efficiently. This method evaluates the function at the midpoint of each subinterval for accurate Riemann sum estimation.