Riemann Sums, the Definite Integral, and the Fundamental Theorem of Calculus
A definite integral is the limit of Riemann sums — it represents net signed area under a curve. The Fundamental Theorem of Calculus connects this accumulation idea back to derivatives and antiderivatives.
Riemann sums approximate area
A Riemann sum approximates ∫[a,b] f(x)dx by slicing [a,b] into n rectangles and summing base×height. Left, right, and midpoint sums use the function value at the left endpoint, right endpoint, or midpoint of each subinterval respectively.
As the number of rectangles n→∞ (width→0), the Riemann sum converges to the exact definite integral — the definite integral IS defined as this limit.
The definite integral gives NET signed area: regions below the x-axis subtract from the total, they don't add positively.
Fundamental Theorem of Calculus (both parts)
FTC Part 1: d/dx [∫ from a to x of f(t)dt] = f(x). In words: differentiating an accumulation function gives back the original integrand (with the chain rule applied if the upper limit is itself a function of x, not just x).
FTC Part 2 (evaluation form): ∫ from a to b of f(x)dx = F(b) - F(a), where F is any antiderivative of f. This is the practical tool for evaluating definite integrals without Riemann sums.
Worked Example
Find d/dx [∫ from 2 to x² of sin(t)dt].
- Let g(x) = ∫ from 2 to u of sin(t)dt where u = x² (upper limit is a function of x, so chain rule is needed).
- By FTC Part 1, d/du[∫ from 2 to u of sin(t)dt] = sin(u).
- Multiply by du/dx = d/dx[x²] = 2x (chain rule).
Answer: d/dx[...] = sin(x²) · 2x
Key terms (2)
- Riemann sum
- A sum of rectangle areas (base × height) used to approximate a definite integral.
- Fundamental Theorem of Calculus
- Links differentiation and integration: accumulation functions differentiate back to the integrand (Part 1); definite integrals evaluate via antiderivatives (Part 2).
Practice Quiz
A left Riemann sum for an increasing function on [a,b] will: