Mathfolis

Definite Integral

Unit 6 · Integration & Accumulation

Definite Integrals and Their Properties

The definite integral ∫ₐᵇ f(x) dx represents the net signed area between f and the x-axis. It satisfies algebraic properties (linearity, additivity, reversal) and, when the integrand is the rate of a quantity, gives the net change of that quantity.

Key Properties

∫aaf(x) dx=0∫abf(x) dx=−∫baf(x) dx\int_a^a f(x)\,dx = 0 \qquad \int_a^b f(x)\,dx = -\int_b^a f(x)\,dx
∫ab[cf±g] dx=c∫abf dx±∫abg dx\int_a^b [cf \pm g]\,dx = c\int_a^b f\,dx \pm \int_a^b g\,dx
∫acf dx+∫cbf dx=∫abf dx\int_a^c f\,dx + \int_c^b f\,dx = \int_a^b f\,dx

Even and Odd Functions on [−a, a]

f even:
∫−aaf(x) dx=2∫0af(x) dx\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx
f odd:
∫−aaf(x) dx=0\int_{-a}^a f(x)\,dx = 0
AP Tip: For g(x) = ∫ₐˣ f(t) dt, g increases when f > 0, decreases when f < 0, and has a local extremum where f changes sign.
Type 1

Applying Properties of Definite Integrals

Use the additive interval property or linearity to compute unknown integrals from given values.

Example 1
Given ∫₀⁶ f(x) dx = 10 and ∫₄⁶ f(x) dx = 3, find ∫₀⁴ f(x) dx.
Example 2
Evaluate ∫₋₃³ (x⁴ − 2x² + 1) dx using symmetry.

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Type 2

Accumulation Function Behavior

Analyze g(x) = ∫ₐˣ f(t) dt by reading the sign and behavior of f to determine where g increases, decreases, or is concave up/down.

Example 3
g(x) = ∫₀ˣ f(t) dt, where f > 0 on (0,2), f(2) = 0, f < 0 on (2,5). (a) Where is g increasing? (b) Classify x = 2. (c) Where is g concave down?

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Type 3

Net vs. Total Area

The definite integral gives net signed area. To find total area, split at x-intercepts and sum |∫| on each piece.

Example 4
The graph of f consists of a triangle above the x-axis on [0, 3] (height 2) and a triangle below on [3, 5] (height 1). Find ∫₀⁵ f(x) dx and the total area.

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