Does more partitions mean more accuracy?

Yes. As n increases, the Riemann sum converges to the exact integral value. With n≥100, most functions yield a highly accurate approximation.

Which method is most accurate?

Among the four methods, the trapezoid rule is generally most accurate, with error O(h²). The midpoint rule also performs well and sometimes outperforms the trapezoid rule.

Why does ln(x) require a positive lower bound?

ln(x) is only defined for x > 0. If the lower bound a ≤ 0, the function is undefined at those points and the calculator will show an error.

∫Definite Integral Approximator

Calculate Riemann sum approximation of definite integral

Select function

Lower bound a

Upper bound b

Partitions n (2 to 1000)

How to Use the Integral Approximator

Select a function, enter the interval [a, b] and number of partitions n to numerically approximate the definite integral using four methods. The larger n is, the closer the result is to the exact integral value.

Four Numerical Integration Methods

Left Riemann Sum: uses the function value at the left endpoint of each subinterval
Right Riemann Sum: uses the function value at the right endpoint
Midpoint Rule: uses the midpoint — generally more accurate than left/right sums
Trapezoid Rule: averages the two endpoints — error O(h²)

Frequently Asked Questions

How many partitions should I use?

For most functions, n=100 gives a good approximation. Increasing to n=1000 makes the result nearly indistinguishable from the exact value.

Why does the left sum overestimate for increasing functions?

For monotonically increasing functions, the left endpoint is always below the curve, so the left Riemann sum underestimates. The right sum overestimates for the same reason.

What is the exact integral of x² from 0 to 2?

∫₀² x² dx = [x³/3]₀² = 8/3 ≈ 2.666667. Try n=100 with this function to verify the approximation accuracy.

How to Use the Integral Approximator

Four Numerical Integration Methods

Frequently Asked Questions

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