Adding It All Up: Integrals
Integration is the reverse of differentiation — it finds the total accumulation of a quantity from its rate of change. The integral of a function represents the area under its curve, connecting it to distance traveled, total revenue earned, volume filled, and countless other accumulation problems. Mastering basic integration rules is essential for all of advanced mathematics.
Components of Integrals
This section covers the fundamental integration techniques:
- Indefinite Integrals: The antiderivative F(x) + C, where C is the constant of integration. The result is a family of functions.
- Power Rule for Integration: The integral of xⁿ is xⁿ⁺¹/(n+1) + C, valid for all n ≠ -1.
- Definite Integrals: The integral from a to b gives a specific number representing the net area under the curve between x = a and x = b.
- Basic Integration Properties: The integral of a sum is the sum of integrals; constants can be factored out.
Examples of Integrals
Indefinite Integral Examples
- Integrate ∫ x³ dx: Using the power rule, x⁴/4 + C.
- Integrate ∫ 5x² dx: Factor out 5, then 5 × x³/3 + C = 5x³/3 + C.
- Integrate ∫ (4x + 3) dx: Integrate term by term to get 2x² + 3x + C.
Power Rule Examples
- Integrate ∫ x⁵ dx = x⁶/6 + C.
- Integrate ∫ √x dx = ∫ x^(1/2) dx = x^(3/2)/(3/2) + C = 2x^(3/2)/3 + C.
- Integrate ∫ 1/x² dx = ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C.
Definite Integral Examples
- Evaluate ∫ from 0 to 3 of 2x dx: The antiderivative is x². Evaluate: 3² - 0² = 9.
- Evaluate ∫ from 1 to 4 of x² dx: Antiderivative is x³/3. Evaluate: 64/3 - 1/3 = 63/3 = 21.
- Find the area under y = 3 from x = 0 to x = 5: ∫ 3 dx = 3x, evaluated from 0 to 5 gives 15 (a rectangle 5 wide and 3 tall).
Integration Properties Examples
- Integrate ∫ (x² + 3x - 1) dx = x³/3 + 3x²/2 - x + C.
- Integrate ∫ 7 × x⁴ dx = 7 × x⁵/5 + C = 7x⁵/5 + C.
- A car travels at v(t) = 3t² m/s. The total distance from t = 0 to t = 4 is ∫ 3t² dt = t³, evaluated from 0 to 4 gives 64 meters.