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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:

Upper Limit:

Lower Limit:

Number of rectangles:

Related

Our online midpoint rule calculator estimates definite integrals using the midpoint method. It approximates the area under a curve by evaluating the function at the midpoint of each subinterval, giving a more accurate estimate compared to simple left- or right-endpoint sums.

What is the Midpoint Rule?

The midpoint rule is a numerical integration technique. It approximates the area under a curve $f(x)$ on an interval [a,b] by summing the areas of rectangles whose heights are determined by the function value at the midpoint of each subinterval.

Mid Point Rule Calculator

You can also try the online Riemann Sum Calculator to compare left, right, midpoint, or trapezoidal estimates.

Midpoint Rule Formula

For n subintervals, the midpoint rule is:

$$ \int_a^b f(x) \, dx \approx \Delta x \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) $$

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

Example

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

Solution:

Δx = (4-1)/5 = 0.6 → Subintervals: 1, 1.6, 2.2, 2.8, 3.4, 4.

Midpoints and function values:

f((1+1.6)/2) = f(1.3) ≈ 2.3853
f((1.6+2.2)/2) = f(1.9) ≈ 2.7586
f((2.2+2.8)/2) = f(2.5) ≈ 3.2015
f((2.8+3.4)/2) = f(3.1) ≈ 3.6891
f((3.4+4)/2) = f(3.7) ≈ 4.2059

Approximate integral:

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

For exact evaluation, use an online integral calculator.

Midpoint Rule Error Bound

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 approximation
n = number of subintervals
K ≥ max |f''(x)| on [a,b]

How the Midpoint Rule Calculator Works

Input:

Enter the function f(x) and limits a and b
Specify the number of subintervals n
Click "Calculate"

Output:

Calculates Δx and midpoints
Shows step-by-step evaluation of f(x) at midpoints
Provides the approximate area under the curve

FAQs

Is the midpoint rule more accurate than the trapezoidal rule?

Generally, yes. The midpoint rule often provides better accuracy for smooth functions, as it samples the function at the center of each subinterval rather than the endpoints.

Why is the trapezoidal rule less accurate?

The trapezoidal rule uses linear approximations, which can underestimate or overestimate areas depending on curvature, while the midpoint rule evaluates at the subinterval center, improving accuracy.

How do you determine midpoints in a Riemann sum?

For each subinterval [x_i, x_{i+1}], the midpoint is x = (x_i + x_{i+1})/2. Evaluate f(x) at these points to set rectangle heights.

How is the error bound calculated?

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

Conclusion

The midpoint rule calculator is a fast and reliable tool for estimating definite integrals. By evaluating the function at the midpoint of each subinterval, it produces accurate Riemann sum approximations for a variety of functions.

References

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