unit-6 – HighFiveAP

6.1 - 6.2

Riemann Sums & Area Approximation

Before we can find exact area using integrals, we approximate it using rectangles (LRAM, RRAM, MRAM) or trapezoids.

Over vs. Under Estimate (Logic)

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.

Function f(x) Left rectangles (LRAM)

For an increasing function, LRAM tends to be an underestimate and RRAM tends to be an overestimate.

Trapezoidal Rule: Concavity decides Over/Under (Graph)

One-Line Memory Hack

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.

Concave Up (∪) → Trapezoid OVERestimate

Concave up means the curve bends upward, so the straight chord sits above it → trapezoids overshoot area.

Concave Down (∩) → Trapezoid UNDERestimate

Concave down means the curve arches over the chord, so trapezoids miss some area → underestimate.

AP Exam Tip: For trapezoids, increasing/decreasing doesn’t decide over/under. Concavity decides it.

6.3

The Definite Integral

The Definite Integral is the limit of a Riemann Sum as the number of rectangles goes to infinity (n → ∞).

∫_a^b f(x) dx = lim_n→∞ Σ f(c_i) Δx

Signed Area

Area above x-axis is Positive (+).

Area below x-axis is Negative (-).

Properties

∫_a^a f(x) dx = 0

∫_a^b = -∫_b^a

6.4

FTC Part 1: Accumulation Functions

This theorem establishes that Differentiation and Integration are inverse operations. It deals with functions defined by integrals.

The Theorem

If g(x) = ∫_a^x f(t) dt, then:

g'(x) = f(x)

"The derivative of the integral is the original function."

Chain Rule Alert: If the upper limit is not just x, but a function u(x):
d/dx ∫_a^u f(t) dt = f(u) · u'
Example: d/dx ∫₂^x³ cos(t) dt = cos(x³) · 3x²

6.5

Analyzing Functions Defined by Integrals

The AP Exam loves to give you the graph of f(t) and ask you questions about g(x) = ∫_a^x f(t) dt.

The "Translation" Key

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.

6.6

Properties of Definite Integrals

You can manipulate integral limits and integrands using these algebraic properties.

Property	Formula
Zero Width	∫_a^a f(x) dx = 0
Reversing Limits	∫_a^b f(x) dx = -∫_b^a f(x) dx
Additivity	∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx
Constant Multiple	∫ k·f(x) dx = k·∫ f(x) dx

6.7

FTC Part 2: Evaluation

This is how we actually calculate the value of a definite integral without using Riemann Sums.

The "Evaluation Bridge"

If F is the antiderivative of f, then:

∫_a^b f(x) dx = F(b) - F(a)

"End Value minus Start Value"

6.8

Indefinite Integrals & Antiderivatives

An indefinite integral ∫ f(x) dx asks for the general family of functions.

DON'T FORGET THE + C
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

6.9

Integration by Substitution (U-Sub)

U-Sub is the "Reverse Chain Rule." It is used when you have a composite function inside the integral.

The U-Sub Algorithm

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.

Golden Rule of Definite U-Sub

Never return to x if you changed bounds to u.

6.10

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)

6.11

Integration by Parts BC Only

Integration by Parts is the reverse of the Product Rule. It is used for products like x e^x or x\sin x.

The Master Formula

∫ u dv = uv - ∫ v du

The LIATE Rule

Logarithmic (ln x)
Inverse Trig
Algebraic
Trig
Exponential

Pick u as the earliest LIATE type.

The Tabular Method

Best when u is a polynomial and dv is easy (like e^x).

Differentiate u until 0.
Integrate dv repeatedly.
Diagonal multiply with alternating signs.

6.12

Linear Partial Fractions BC Only

Decompose rational functions into simpler pieces so you can integrate with ln|x| rules.

The "Cover-Up" Method

Problem: Evaluate ∫ (5x - 1) / ((x-1)(x+3)) dx

Setup: (5x-1)/((x-1)(x+3)) = A/(x-1) + B/(x+3)
Find A: plug x=1 → A=1
Find B: plug x=-3 → B=4
Integrate: ln|x-1| + 4ln|x+3| + C

6.13

Improper Integrals BC Only

Improper Integrals involve ∞ bounds or vertical asymptotes. You must use limits.

Mandatory FRQ Writing Format:

❌ WRONG: ∫₁^∞ (1/x²) dx = [-1/x]₁^∞

✅ RIGHT: lim_b→∞ ∫₁^b (1/x²) dx

The P-Test Shortcut (MCQ)

For ∫₁^∞ 1/xᵖ dx:

If p > 1, it converges.
If p ≤ 1, it diverges.

6.14

Selecting Techniques for Integration (BC)

When faced with a random integral on the BC exam, walk through this Decision Tree in your head:

Is it a basic rule? Power rules, 1/x, eˣ, trig.
Can I use U-Substitution? Inside function + its derivative nearby.
Is it a product? Use Integration by Parts BC.
Is it a Rational Function?
- Top-heavy → Long Division.
- Unfactorable quadratic → Complete the Square (arctan form).
- Factorable linears → Partial Fractions BC.
Infinity or asymptote? Use Improper Integral BC with limits.