免费黎曼和计算器
使用黎曼和近似计算定积分，支持左端点和右端点法。
Exact Val
0.333333
公式
## Riemann Sums ### Concept A Riemann sum approximates the area under a curve by dividing the interval into rectangles: 1. Divide [a, b] into n equal subintervals, each of width Δx = (b-a)/n 2. For each subinterval, evaluate the function at the left endpoint 3. Sum the areas: S = sum of f(xᵢ) × Δx ### As n → infinity The Riemann sum converges to the exact definite integral. More rectangles = better approximation.
计算示例
Approximate the integral of x² from 0 to 1 with 10 rectangles.
- 01Δx = (1-0)/10 = 0.1
- 02Left endpoints: 0, 0.1, 0.2, ..., 0.9
- 03Sum = (0² + 0.1² + 0.2² + ... + 0.9²) × 0.1
- 04Exact value = 1³/3 = 0.333333...
常见问题
What is a Riemann sum?
A Riemann sum is a method for approximating the area under a curve by dividing it into rectangles and summing their areas. It is the foundation for the definite integral.
Do more rectangles give a better answer?
Yes, as the number of rectangles increases, the approximation gets closer to the exact value. In the limit as n → infinity, it equals the integral.
What are left, right, and midpoint sums?
Left sums evaluate at the left endpoint of each subinterval, right sums at the right, and midpoint sums at the center. Midpoint sums are generally the most accurate.
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