Definite integrals measure accumulated change over an interval, often shown as signed area between a graph and the x-axis. They are central in physics, engineering, economics, and geometry because many quantities are built by adding tiny pieces. The main properties of definite integrals let you rewrite, combine, estimate, and simplify integrals without always finding an antiderivative.
These rules make complicated area and accumulation problems easier to organize.
Key Facts
- Additivity over intervals: ∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx.
- Reversing limits changes the sign: ∫_a^b f(x) dx = -∫_b^a f(x) dx.
- Same limits give zero: ∫_a^a f(x) dx = 0.
- Constant multiple rule: ∫_a^b kf(x) dx = k∫_a^b f(x) dx.
- Sum and difference rule: ∫_a^b [f(x) ± g(x)] dx = ∫_a^b f(x) dx ± ∫_a^b g(x) dx.
- Bounds property: if m ≤ f(x) ≤ M on [a, b], then m(b - a) ≤ ∫_a^b f(x) dx ≤ M(b - a).
Vocabulary
- Definite integral
- A definite integral ∫_a^b f(x) dx gives the signed accumulation of f(x) from x = a to x = b.
- Signed area
- Signed area counts regions above the x-axis as positive and regions below the x-axis as negative.
- Limits of integration
- The limits of integration are the lower and upper x-values that define the interval of accumulation.
- Additivity
- Additivity is the property that an integral over a long interval can be split into integrals over smaller adjacent intervals.
- Bounds
- Bounds are upper and lower limits on a function that can be used to estimate the possible value of its definite integral.
Common Mistakes to Avoid
- Ignoring the sign of area below the x-axis. A definite integral uses signed area, so a region below the x-axis contributes a negative value.
- Forgetting to change the sign when reversing limits. The integral ∫_b^a f(x) dx is the opposite of ∫_a^b f(x) dx, not the same value.
- Splitting intervals in the wrong order. Additivity works when the subintervals connect properly, such as from a to b and b to c.
- Applying bounds without checking the whole interval. The inequalities m ≤ f(x) ≤ M must hold for every x in [a, b], not just at the endpoints.
Practice Questions
- 1 If ∫_1^4 f(x) dx = 9 and ∫_4^7 f(x) dx = -2, find ∫_1^7 f(x) dx.
- 2 If ∫_2^6 g(x) dx = 5, find ∫_6^2 3g(x) dx.
- 3 A graph of f(x) has equal geometric areas above and below the x-axis on [0, 8]. Explain whether ∫_0^8 f(x) dx must be zero, and state what extra information would be needed if it is not clear.