Riemann Sum Calculator
Numerical Integration & Definite Integral Approximation
Calculate Riemann sums to approximate definite integrals using left, right, midpoint, and trapezoidal methods with detailed step-by-step analysis.
Numerical integration using Riemann sums
∫02 (x^2) dx ≈ Σi=110 f(xi) · Δx
Left Rule
Uses the function value at the left endpoint of each subinterval. Good for decreasing functions.
Right Rule
Uses the function value at the right endpoint of each subinterval. Good for increasing functions.
Midpoint Rule
Uses the function value at the midpoint of each subinterval. Generally more accurate than left/right rules.
Trapezoidal Rule
Uses trapezoids instead of rectangles. Often the most accurate method for the same number of subintervals.
Integration Methods
Riemann Sum Methods
- • Left Rule: Uses left endpoints of subintervals
- • Right Rule: Uses right endpoints of subintervals
- • Midpoint Rule: Uses midpoints of subintervals
- • Trapezoidal Rule: Uses trapezoids for better accuracy
Supported Functions
- • Polynomial functions: x², x³, etc.
- • Trigonometric: sin(x), cos(x), tan(x)
- • Exponential: e^x, 2^x
- • Logarithmic: ln(x), log(x)
- • Square root: sqrt(x)
- • Custom expressions: x²+2x+1
Frequently Asked Questions
What is a Riemann sum?
A Riemann sum is a method for approximating the definite integral of a function by dividing the area under the curve into rectangles and summing their areas.
Which method is most accurate?
The trapezoidal rule is generally more accurate than left/right/midpoint rules for the same number of subintervals. For even better accuracy, increase the number of subintervals.
How many subintervals should I use?
More subintervals generally give better approximations, but take longer to compute. Start with 10-50 for quick estimates, use 100-1000 for higher precision.
Can I visualize the rectangles?
Yes! Our calculator shows a table of all rectangle calculations, including the x-value, function value, and area of each subinterval for complete transparency.
What's the difference between left and right Riemann sums?
Left Riemann sums use the function value at the left endpoint of each subinterval, while right sums use the right endpoint. This can lead to overestimation or underestimation depending on whether the function is increasing or decreasing.
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