Riemann Sums & Area Approximation
Before we can find exact area using integrals, we approximate it using rectangles (LRAM, RRAM, MRAM) or trapezoids.
You don't need to draw it every time. Memorize the logic:
| Function Behavior | LRAM (Left) | RRAM (Right) |
|---|---|---|
| Increasing (↗) | Under Estimate | Over Estimate |
| Decreasing (↘) | Over Estimate | Under Estimate |
Note: Trapezoidal Sum depends on Concavity. Concave Up = Over; Concave Down = Under.
Trapezoids use chords (straight lines). If the curve is concave up, chords lie above the curve → trapezoids OVER. If the curve is concave down, chords lie below the curve → trapezoids UNDER.
The Definite Integral
The Definite Integral is the limit of a Riemann Sum as the number of rectangles goes to infinity (n → ∞).
Area above x-axis is Positive (+).
Area below x-axis is Negative (-).
∫aa f(x) dx = 0
∫ab = -∫ba
FTC Part 1: Accumulation Functions
This theorem establishes that Differentiation and Integration are inverse operations. It deals with functions defined by integrals.
If g(x) = ∫ax f(t) dt, then:
"The derivative of the integral is the original function."
d/dx ∫au f(t) dt = f(u) · u'
Example: d/dx ∫2x³ cos(t) dt = cos(x³) · 3x²
Analyzing Functions Defined by Integrals
The AP Exam loves to give you the graph of f(t) and ask you questions about g(x) = ∫ax f(t) dt.
Because g'(x) = f(x), the graph of f is the derivative graph of g.
- If f > 0, then g is increasing.
- If f < 0, then g is decreasing.
- If f is increasing (f' > 0), then g is concave up.
- If f is decreasing (f' < 0), then g is concave down.
Properties of Definite Integrals
You can manipulate integral limits and integrands using these algebraic properties.
| Property | Formula |
|---|---|
| Zero Width | ∫aa f(x) dx = 0 |
| Reversing Limits | ∫ab f(x) dx = -∫ba f(x) dx |
| Additivity | ∫ab f(x) dx + ∫bc f(x) dx = ∫ac f(x) dx |
| Constant Multiple | ∫ k·f(x) dx = k·∫ f(x) dx |
FTC Part 2: Evaluation
This is how we actually calculate the value of a definite integral without using Riemann Sums.
If F is the antiderivative of f, then:
"End Value minus Start Value"
Indefinite Integrals & Antiderivatives
An indefinite integral ∫ f(x) dx asks for the general family of functions.
If you write the antiderivative without + C, you will lose the point.
| Rule | Integral Formula |
|---|---|
| Reverse Power | ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C |
| 1/x Rule | ∫ (1/x) dx = ln|x| + C |
| Exponential | ∫ eˣ dx = eˣ + C |
| Trig | ∫ cos(x) dx = sin(x) + C ∫ sin(x) dx = -cos(x) + C |
Integration by Substitution (U-Sub)
U-Sub is the "Reverse Chain Rule." It is used when you have a composite function inside the integral.
Problem: Evaluate ∫01 2x(x² + 1)³ dx
- Choose u: Let u = x² + 1.
- Differentiate: du = 2x dx.
- Change Limits: x=0→u=1, x=1→u=2.
- Integrate: ∫12 u³ du = [u⁴/4]12 = 15/4.
Never return to x if you changed bounds to u.
Algebraic Manipulation (Division & Completing the Square)
When basic integration and U-Substitution fail, we must alter the algebraic form of the integrand before integrating.
Long Division
When to use: Top-heavy rational functions (deg numerator ≥ deg denominator).
Example: ∫ (x² + x)/x dx = ∫ (x + 1) dx
Completing the Square
When to use: Irreducible quadratic → aim for (x+a)² + b².
Example: 1/(x²+2x+2) = 1/((x+1)²+1)
Selecting Techniques for Integration
When faced with a random integral on the exam, walk through this Decision Tree in your head to avoid getting stuck:
- Is it a basic rule? Look for simple power rules, 1/x, e^x, or basic trig functions. Expand or simplify first if needed.
- Can I use U-Substitution? Look for an "inside" function whose derivative is floating around on the "outside" (e.g., 2x outside of sin(x²)).
- Is it a Rational Function (Fractions)?
- Is the numerator degree $\ge$ denominator degree? Use Long Division.
- Is the denominator an unfactorable quadratic? Complete the Square (leads to arctan).