Integrals · Fundamental Theorems
Riemann Sum
The definite integral is defined as the limit of Riemann sums. Partition [a,b] into n subintervals of width Δx = (b-a)/n.
- 3 variables
- 2 worked examples
- 19 in Integrals
Variables
| Symbol | Name | Unit |
|---|---|---|
| a | Lower bound | - |
| b | Upper bound | - |
| n | Number of subintervals | - |
Worked examples
Approximate ∫₀¹ x² dx using a right Riemann sum with n = 4.
- Δx = 1/4. Right endpoints: 1/4, 1/2, 3/4, 1
- Sum = (1/16 + 1/4 + 9/16 + 1)(1/4) = (1/16 + 4/16 + 9/16 + 16/16)(1/4)
- = (30/16)(1/4) = 30/64 = 15/32
Answer: 15/32 = 0.46875 (exact: 1/3 ≈ 0.3333)
1 more worked examplePremium
Approximate ∫₀¹ x² dx using a left Riemann sum with n = 4.
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Practice
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