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.
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. You must use FTC Part 1 to translate between them.
Because g'(x) = f(x), the graph of f is actually the derivative graph of g.
- If the graph of f is above the x-axis (f > 0), then g is increasing.
- If the graph of f is below the x-axis (f < 0), then g is decreasing.
- If the graph of f is increasing (f' > 0), then g is concave up.
- If the graph of 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 (Flipping limits flips the sign!) |
| 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 (meaning F' = 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. Every time. No exceptions.
| Rule | Integral Formula |
|---|---|
| Reverse Power | ∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C |
| 1/x Rule | ∫ (1/x) dx = ln|x| + C (Absolute value is crucial!) |
| 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 (the inside function).
- Differentiate: du = 2x dx → dx = du / 2x.
- Change Limits:
- If x = 0, u = 0² + 1 = 1.
- If x = 1, u = 1² + 1 = 2.
- Substitute: ∫12 u³ du (The 2x cancels out).
- Integrate: [u⁴/4] from 1 to 2.
- Evaluate: (2⁴/4) - (1⁴/4) = 4 - 0.25 = 3.75.
Never return to x if you have changed your bounds to u. Just finish the problem in u-world. It's faster and less prone to error.
Algebraic Manipulation (Division & Completing the Square)
When basic integration and U-Substitution fail, we must alter the algebraic form of the integrand before integrating. These two algebra tricks are essential for AB Calculus.
Long Division
When to use: When the fraction is "Top-Heavy" (the degree of the numerator is $\ge$ the degree of the denominator).
How: Use polynomial long division to rewrite the improper fraction as a polynomial plus a proper fraction.
Example: ∫ (x² + x) / x dx = ∫ (x + 1) dx
Completing the Square
When to use: When the denominator is an irreducible quadratic (like $x^2 + 4x + 5$) and the numerator is a constant.
How: Force the denominator into the form (x+a)² + b², which perfectly sets up the arctan integration formula.
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).