The definite integral is one of the central ideas of calculus because it turns continuous change into a total amount. It is used to find areas, distances, accumulated quantities, work, probability, and many other totals. On a graph, it connects algebraic notation with a visual idea: the region between a curve and the x-axis over an interval.
Understanding it helps students move from local rates of change to whole-interval behavior.
A definite integral is defined as the limit of Riemann sums, where many thin rectangles approximate the area under a curve. As the rectangle widths approach zero, the approximation approaches an exact signed area. Regions above the x-axis contribute positive area, while regions below the x-axis contribute negative area.
The notation ∫_a^b f(x) dx records the function being accumulated, the interval of accumulation, and the variable whose small changes are being added.
Key Facts
- Definite integral notation: ∫_a^b f(x) dx
- Riemann sum form: ∑ from i = 1 to n of f(x_i*) Δx
- Equal subinterval width: Δx = (b - a)/n
- Definite integral as a limit: ∫_a^b f(x) dx = lim as n approaches infinity of ∑ from i = 1 to n f(x_i*) Δx
- Signed area rule: area above the x-axis is positive and area below the x-axis is negative.
- Fundamental Theorem of Calculus: if F'(x) = f(x), then ∫_a^b f(x) dx = F(b) - F(a).
Vocabulary
- Definite integral
- A number that represents the accumulated signed area of a function over a specific interval.
- Riemann sum
- A sum of rectangle areas used to approximate a definite integral.
- Signed area
- Area counted as positive when the graph is above the x-axis and negative when the graph is below the x-axis.
- Subinterval
- One smaller piece of the interval from a to b used when building a Riemann sum.
- Antiderivative
- A function F whose derivative is the original function f.
Common Mistakes to Avoid
- Forgetting that area can be negative, which is wrong because a definite integral measures signed area, not always total geometric area.
- Treating dx as decoration, which is wrong because it tells the variable of integration and represents the limiting width of the rectangles.
- Using the wrong interval endpoints, which is wrong because ∫_a^b f(x) dx accumulates only from x = a to x = b.
- Confusing a definite integral with an indefinite integral, which is wrong because a definite integral gives a number while an indefinite integral gives a family of functions.
Practice Questions
- 1 Compute ∫_0^3 2x dx using geometry or an antiderivative.
- 2 Approximate ∫_0^4 x^2 dx using 4 equal right-endpoint rectangles.
- 3 A function is positive on [0, 2] and negative on [2, 5]. Explain why ∫_0^5 f(x) dx could be zero even if the graph encloses visible area.