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Definite Integrals

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Definite Integrals - Complete Interactive Lesson

Part 1: Riemann Sums

∫ Definite Integrals

Part 1 of 7 — Riemann Sums

Part	Topic
1	Riemann Sums
2	Definite Integral Definition
3	Properties of Integrals
4	FTC Part 1
5	FTC Part 2 & Net Change
6	Problem-Solving Workshop
7	Comprehensive Review

The Area Problem

How do we find the exact area under a curve? Approximate with rectangles, then take $n \to \infty$ .

$\boxed{\Delta x = \frac{b - a}{n}}$

Left, Right, and Midpoint Sums

Type	Sample Point	Formula
Left ( $L_n$ )	Left endpoint $x_i$	$\sum_{i =}^{}$

Worked Example

Approximate $\int_0^4 x^2\,dx$ with $n = 4$ using Left Riemann Sum.

$\Delta x = 1$ . Left endpoints: $0, 1, 2, 3$ .

$L_4 = f(0) + f(1) + f(2) + f(3) = 0 + 1 + 4 + 9 = 14$

The exact answer is $\frac{64}{3} \approx 21.33$ , so $L_4 = 14$ is an .

Over- and Underestimates — Essential AP Skill

$\boxed{\text{The relationship between the function's behavior and the estimate type determines over/under.}}$

Function Behavior	Left Sum	Right Sum	Midpoint	Trapezoidal
Increasing	Under	Over	—	Over
Decreasing	Over	Under	—	Under

Riemann Sum Computations 🎯

Trapezoidal Rule

The Trapezoidal Rule averages the Left and Right sums:

$\boxed{T_n = \frac{\Delta x}{2}\left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]}$

Trapezoidal Rule from a Table 🎯

Given the table:

$x$	0	2	5	8	10
$f(x)$	3	7	11	6	4

Classify each approximation. 🔍

Let $f$ be a positive, increasing, concave-up function on $[a,b]$ .

Compute a Riemann Sum. ✍️

Key Takeaways — Part 1

Concept	Key Formula
Subinterval width	$\Delta x = (b-a)/n$
Left Sum	Use left endpoints
Right Sum	Use right endpoints
Midpoint Sum	Use midpoint of each subinterval
Trapezoidal	$\frac{Δ x}{2} [f (_{}$

Part 2: Definite Integral Definition

∫ Definite Integrals

Part 2 of 7 — The Definite Integral

From Riemann Sums to Exact Area

The definite integral is the limit of a Riemann sum as $n \to \infty$ :

$\boxed{\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x}$

Part 3: Properties of Integrals

∫ Definite Integrals

Part 3 of 7 — Properties of Integrals

Essential Properties

$\boxed{\int_a^b [cf(x)]\,dx = c\int_a^b f(x)\,dx}$

Part 4: Fundamental Theorem of Calculus — Part 1

∫ Definite Integrals

Part 4 of 7 — FTC Part 1

The Fundamental Theorem — Part 1

$\boxed{\frac{d}{dx}\int_a^x f(t)\,dt = f(x)}$

Part 5: Fundamental Theorem of Calculus — Part 2

∫ Definite Integrals

Part 5 of 7 — FTC Part 2 & Net Change

The Evaluation Theorem (FTC Part 2)

$\boxed{\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where } F' = f}$

Part 6: Mixed Integration Problems

∫ Definite Integrals

Part 6 of 7 — Problem-Solving Workshop

Combining All Tools

This part brings together everything: Riemann sums, FTC, properties, and applications.

Strategy Guide

Problem Type	Key Approach
Evaluate $\int_a^b f(x)\,dx$	Find antiderivative, apply FTC Part 2

Part 7: Comprehensive Review

∫ Definite Integrals — Comprehensive Review

Part 7 of 7 — Final Assessment

Complete Summary

Concept	Key Formula
Riemann Sum	$\sum f(x_i^*) \Delta x$
Trapezoidal Rule	$\frac{}{}$

\sum_{i = 0}^{n - 1} f (x_{i}) Δ x

Key Concept: For increasing functions, left rectangles miss the top-right corner (under), while right rectangles include extra area (over). The reverse for decreasing.

AP Tip: The AP Exam loves asking "Is this an over- or underestimate?" You MUST justify by citing whether $f$ is increasing/decreasing (for L/R) or concave up/down (for M/T).

[f(x0)+2f(x1)+2f(x2)+⋯+2f(xn−1)+f(xn)]

Pattern: First and last values appear ONCE; all middle values are DOUBLED.

Equal vs. Unequal Subintervals

With unequal subintervals (common on AP tables), apply the trapezoid formula to each pair:

$T = \frac{\Delta x_1}{2}[f(x_0) + f(x_1)] + \frac{\Delta x_2}{2}[f(x_1) + f(x_2)] + \cdots$

Trapezoidal approximation of $\int_0^4 x^2\,dx$ with $n = 4$ :

$T_4 = \frac{L_4 + R_4}{2} = \frac{14 + 30}{2} = 22$

Exact: $64/3 \approx 21.33$ . Since $x^2$ is concave up, the Trapezoidal sum overestimates. ✓

AP Tip: The trapezoidal rule with table data is one of the most common AP FRQ questions. Practice with unequal subintervals!

\frac{Δ x}{2} [f (x_{0}) + 2 f (x_{1}) + \dots + f (x_{n})]

Up Next: Part 2 — The Definite Integral.

∫abf(x)dx=n→∞limi=1∑nf(xi∗)Δx

Region	Sign
Between curve and $x$ -axis, curve ABOVE axis	Positive
Between curve and $x$ -axis, curve BELOW axis	Negative

$\boxed{\text{Signed area} = \int_a^b f(x)\,dx \qquad \text{Total area} = \int_a^b |f(x)|\,dx}$

Key Concept: The definite integral gives signed area, not total area. The AP Exam tests this distinction frequently!

Geometric Evaluation

Some integrals can be evaluated using geometry instead of antiderivatives:

Shape	Integral	Value
Rectangle	$\int_0^3 5\,dx$	$5 \times 3 = 15$
Triangle	$\int_0^4 2x\,dx$	$\frac{1}{2}(4)(8) = 16$
Trapezoid	$\int_0^3 (2x+1)\,dx$	$\frac{1}{2}(1+7)(3) = 12$
Semicircle	$\int_{-r}^{r} \sqrt{r^2-x^2}\,dx$

Verify: $\int_0^3 (2x+1)\,dx = [x^2 + x]_0^3 = 12$ ✓

Worked Example — Semicircle

$\int_{-3}^{3} \sqrt{9-x^2}\,dx$ = area of semicircle with =

AP Tip: If you see $\sqrt{r^2 - x^2}$ , think semicircle! No antiderivative needed.

Definite Integral Concepts 🎯

Even and Odd Function Shortcuts

$\boxed{\text{Odd: } \int_{-a}^{a} f(x)\,dx = 0 \qquad \text{Even: } \int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx}$

Type	Definition	Examples	Integral on $[-a, a]$
Odd	$f(-x) = -f(x)$

Worked Example

$\int_{-1}^{1} (x^4 + x^3)\,dx$

Split: $\underbrace{\int_{-1}^1 x^4\,dx}_{\text{even}} + \underbrace{\int_{-1}^1 x^3\,dx}_{\text{odd}}$

$= 2\int_0^1 x^4\,dx + 0 = 2 \cdot \frac{1}{5} = \frac{2}{5}$

AP Tip: Check for symmetry BEFORE computing! It can save significant time on the exam.

Evaluate Definite Integrals 🎯

Interpret each integral. 🔍

Geometric Evaluation ✍️

Key Takeaways — Part 2

Concept	Key Point
Definition	$\int_a^b f\,dx = \lim_{n \to \infty} \sum f(x_i^*)\Delta x$
Signed area	Above axis = +, below = −
Total area	$\int
Odd functions	Integral = 0 on symmetric intervals
Even functions	$2 \times$ integral from 0 to $a$
Geometry	Use triangles, trapezoids, semicircles

Up Next: Part 3 — Properties of Integrals.

$\boxed{\int_a^b [f(x) \pm g(x)]\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx}$

Property	Formula	Interpretation
Constant Multiple	$\int_a^b cf\,dx = c\int_a^b f\,dx$	Factor constants out
Sum/Difference	$\int_a^b (f \pm g)\,dx = \int f \pm \int g$
Additivity	$\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx$
Reversal	$\int_a^b f\,dx = -\int_b^a f\,dx$
Zero Width	$\int_a^a f\,dx = 0$	No interval = no area
Comparison	$f \geq g \Rightarrow \int f \geq \int g$	Bigger function = bigger integral

Key Concept: Integrals are linear — they respect addition and scalar multiplication. This is used constantly on the AP Exam!

Worked Examples — Given-Value Problems

A common AP question gives you known integral values and asks you to find others.

Given: $\int_0^5 f(x)\,dx = 10$ , $\int_0^5 g(x)\,dx = 3$ , $\int_0^3 f(x)\,dx = 7$ .

Find	Work	Answer
$\int_0^5 [2f(x) - 3g(x)]\,dx$

AP Tip: Don't forget that $\int_a^b k\,dx = k(b-a)$ for a constant $k$ . Many students miss the constant term!

Apply Integral Properties 🎯

Given: $\int_0^5 f(x)\,dx = 10$ and $\int_0^5 g(x)\,dx = 3$ .

Average Value of a Function

$\boxed{f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx}$

Interpretation: The average height of the function over $[a,b]$ .

Geometric meaning: $f_{\text{avg}} \cdot (b-a) = \int_a^b f(x)\,dx$ . The rectangle with height has the SAME area as the region under the curve.

Worked Example

Find the average value of $f(x) = x^2$ on $[0, 3]$ .

$f_{\text{avg}} = \frac{1}{3-0}\int_0^3 x^2\,dx = \frac{1}{3} \cdot \left[\frac{x^3}{3}\right]_0^3 = \frac{1}{3} \cdot 9 = 3$

AP Tip: The Mean Value Theorem for Integrals guarantees there exists a $c \in [a,b]$ where $f(c) = f_{\text{avg}}$ (if is continuous). They may ask you to find this .

Average Value 🎯

Apply Properties 🔍

Given: $\int_0^6 f(x)\,dx = 15$ , $\int_0^6 g(x)\,dx = 7$ , $\int_0^4 f(x)\,dx = 9$ .

Compute with Properties ✍️

Key Takeaways — Part 3

Property	Formula
Linearity	Constants factor out, sums split
Additivity	$\int_a^b + \int_b^c = \int_a^c$
Reversal	Swap limits → flip sign
Average value	$\frac{1}{b-a}\int_a^b f\,dx$
Constant integral	$\int_a^b k\,dx = k(b-a)$

Up Next: Part 4 — FTC Part 1.

In words: Differentiation undoes integration. If you integrate $f$ and then differentiate, you get $f$ back.

If the upper limit is a function $g(x)$ :

$\boxed{\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)}$

All Variations at a Glance

Situation	Formula	Key Step
Upper limit = $x$	$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$	Direct application
Upper limit = $g(x)$	$f(g(x)) \cdot g'(x)$	Chain Rule
Lower limit = $x$	$\frac{d}{dx}\int_x^b f(t)\,dt = -f(x)$
Both limits are functions	$f(g(x))g'(x) - f(h(x))h'(x)$

AP Tip: FTC Part 1 with the Chain Rule is tested almost every year on the AP Exam. Master this!

Worked Examples

Example 1: $\frac{d}{dx}\int_2^x t^3\,dt = x^3$ ✓ (direct)

Example 2: $\frac{d}{dx}\int_0^{x^2} \sin(t)\,dt$

$g(x) = x^2$ , $g'(x) = 2x$ :

Example 3: $\frac{d}{dx}\int_x^5 e^{t^2}\,dt$

Reverse: $= -\frac{d}{dx}\int_5^x e^{t^2}\,dt = -e^{x^2}$

Example 4 (Both limits): $\frac{d}{dx}\int_{2x}^{x^3} \cos(t)\,dt$

Split: $\int_0^{x^3} \cos t\,dt - \int_0^{2x} \cos t\,dt$

$= \cos(x^3) \cdot 3x^2 - \cos(2x) \cdot 2 = 3x^2\cos(x^3) - 2\cos(2x)$

Accumulation Functions

$F(x) = \int_a^x f(t)\,dt$ is an accumulation function: it measures how much $f$ has "accumulated" from $a$ to $x$ .

Connecting $f$ and $F$

About $f$	About $F = \int_a^x f\,dt$

$\boxed{F(a) = \int_a^a f(t)\,dt = 0 \qquad \text{(always starts at 0)}}$

Key Concept: If they give you the graph of $f'$ , you can determine the behavior of $f$ using this same table (since $f = \int f'\,dt$ ). This is one of the most common AP graph-analysis questions.

Accumulation Functions 🎯

Let $F(x) = \int_0^x f(t)\,dt$ where $f$ is continuous.

FTC Part 1 — Match the derivative. 🔍

Compute a specific value. ✍️

Key Takeaways — Part 4

Concept	Formula
FTC Part 1 (basic)	$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$
FTC Part 1 (chain)	$\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$
Variable in lower limit	Reverse limits → negative sign
$F(a) = 0$	Accumulation starts at 0
$f > 0 \Rightarrow F$ increasing	$f < 0 \Rightarrow F$ decreasing

Up Next: Part 5 — FTC Part 2 & Net Change.

Notation: $F(x)\Big|_a^b$ or $\left[F(x)\right]_a^b = F(b) - F(a)$

Quick Evaluation Examples

Integral	Antiderivative	Evaluation
$\int_0^2 3x^2\,dx$	$x^3$	$8 - 0 = 8$
$\int_1^e \frac{1}{x}\,dx$	$\ln$
$\int_0^1 e^x\,dx$	$e^x$
$\int_0^{\pi/2} \cos x\,dx$	$\sin x$	$1 - 0 = 1$

Key Fact: You can use ANY antiderivative — so always choose $C = 0$ for simplicity.

Evaluate Using FTC Part 2 🎯

Net Change Theorem

$\boxed{\int_a^b f'(x)\,dx = f(b) - f(a)}$

The integral of a rate of change gives the NET CHANGE in the original quantity.

Applications Table

Quantity	Rate	$\int_a^b (\text{rate})\,dt$ gives...
Position $s(t)$	Velocity

Displacement vs Total Distance

$\boxed{\text{Displacement} = \int_a^b v(t)\,dt \qquad \text{Total Distance} = \int_a^b |v(t)|\,dt}$

Concept	Formula	Includes direction?
Displacement	$\int v\,dt$	Yes (can be negative)
Total Distance	$\int	v

AP Tip: "How far" = total distance ( $\int |v|$ ). "What is the displacement" or "change in position" = $\int v$ . Read the question carefully!

Worked Example — Displacement vs Distance

A particle has $v(t) = t^2 - 4$ on $[0, 3]$ .

Displacement: $\int_0^3 (t^2 - 4)\,dt = [\frac{t^3}{3} - 4t]_0^3 = (9 - 12) - 0 = -3$

The particle is 3 units to the LEFT of where it started.

Total Distance: $v(t) = 0$ at $t = 2$ . Split at the zero:

$\int_0^2 |t^2-4|\,dt + \int_2^3 |t^2-4|\,dt = \int_0^2 (4-t^2)\,dt + \int_2^3 (t^2-4)\,dt$

$= [4t - \frac{t^3}{3}]_0^2 + [\frac{t^3}{3} - 4t]_2^3 = \frac{16}{3} + \frac{7}{3} = \frac{23}{3}$

Key Concept: To compute $\int |v|\,dt$ , find where $v = 0$ , split the integral, and negate $v$ on intervals where $v < 0$ .

Net Change Theorem 🎯

Interpret each integral. 🔍

Net Change Problem ✍️

Key Takeaways — Part 5

Concept	Formula
FTC Part 2	$\int_a^b f = F(b) - F(a)$
Net Change	$\int_a^b f' = f(b) - f(a)$
Displacement	$\int_a^b v\,dt$ (signed)
Total Distance	$\int_a^b
Position update	$s(b) = s(a) + \int_a^b v\,dt$

Up Next: Part 6 — Problem-Solving Workshop.

AP Tip: On FRQs, always show your setup (the integral expression) before evaluating. Setup points are awarded separately from answer points.

AP-Style Mixed Problems — Set 1 🎯

Absolute Value Integrals — Step by Step

To evaluate $\int_a^b |f(x)|\,dx$ :

Find where $f(x) = 0$ (the zeros)
Determine sign of $f$ on each subinterval
Split the integral at each zero
Negate $f$ on intervals where $f < 0$

Worked Example

$\int_0^4 |x - 2|\,dx$

$x - 2 = 0$ at $x = 2$ .

On $[0,2]$ : $x - 2 < 0$ , so $|x-2| = -(x-2) = 2-x$

$= \int_0^2 (2-x)\,dx + \int_2^4 (x-2)\,dx = [2x - \frac{x^2}{2}]_0^2 + [\frac{x^2}{2} - 2x]_2^4$

$= (4 - 2) + (8 - 8) - (2 - 4) = 2 + 0 + 2 = 4$

Geometric shortcut: $|x-2|$ forms a V-shape — two right triangles each with base 2 and height 2. Area = $2 \times \frac{1}{2}(2)(2) = 4$ . ✓

Mixed Problems — Set 2 🎯

Classify each problem type. 🔍

Trapezoidal Rule from a Table ✍️

$t$ (min)	0	3	7	10
$R(t)$ (gal/min)	4	6	10	8

Key Takeaways — Part 6

Problem Type	Go-To Tool
Evaluate definite integral	FTC Part 2
Differentiate an integral	FTC Part 1
Given values problems	Properties (linearity, additivity)
Rate → amount	Net Change Theorem
Table data	Trapezoidal Rule
Absolute value	Split at zeros

Up Next: Part 7 — Comprehensive Review.

\frac{Δ x}{2} [f (x_{0}) + 2 f (x_{1}) + \dots + f (x_{n})]

Common AP Exam Mistakes

Mistake	Consequence
Forgetting Chain Rule on FTC Part 1	Missing the $g'(x)$ factor
Confusing displacement with distance	Using $\int v$ when asked for $\int
Not splitting at zeros for $\int	f
Forgetting $\int k\,dx = k(b-a)$ for constants	Missing the constant term
Wrong Trapezoidal with unequal widths	Using equal $\Delta x$ when widths vary
Not reversing limits when $x$ is in lower bound	Sign error

Final Assessment — Set 1 🎯

Final Assessment — Set 2 🎯

AP-Style Comprehensive 🔍

Final Challenge ✍️

Definite Integrals — Complete! ✅

You have mastered:

Skill	Parts
Riemann Sums (L, R, M, T)	Part 1
Over/Underestimate analysis	Part 1
Signed area & geometry	Part 2
Even/odd symmetry	Part 2
Properties & average value	Part 3
FTC Part 1 (+ Chain Rule)	Part 4
Accumulation functions	Part 4
FTC Part 2 & Net Change	Part 5
Displacement vs distance	Part 5
Mixed problem solving	Parts 6-7

AP Exam Checklist

✅ Can evaluate definite integrals with FTC Part 2
✅ Can differentiate integrals with FTC Part 1 (+ Chain Rule)
✅ Can use properties to compute from given values
✅ Can apply Trapezoidal Rule to table data
✅ Can distinguish displacement from total distance
✅ Can analyze accumulation functions from graphs of $f$

$\boxed{\text{FTC connects differentiation and integration — they are INVERSE operations!}}$

\sin(x^2) \cdot 2x = 2x\sin(x^2)

sin (x^{2}) \cdot 2 x = 2 x sin (x^{2})

(always starts at 0)

[2,4]

x - 2 > 0

, so

|x-2| = x-2

Definite Integrals

Equal vs. Unequal Subintervals

Worked Example

Signed Area

Geometric Evaluation

Worked Example — Semicircle

Even and Odd Function Shortcuts

Worked Example

Key Takeaways — Part 2

Worked Examples — Given-Value Problems

Average Value of a Function

Worked Example

Key Takeaways — Part 3

With the Chain Rule

All Variations at a Glance

Worked Examples

Accumulation Functions

Connecting $f$ and $F$

Key Takeaways — Part 4

Quick Evaluation Examples

Net Change Theorem

Applications Table

Displacement vs Total Distance

Worked Example — Displacement vs Distance

Key Takeaways — Part 5

Absolute Value Integrals — Step by Step

Worked Example

Key Takeaways — Part 6

Common AP Exam Mistakes

Definite Integrals — Complete! ✅

AP Exam Checklist

Definite Integrals

Definite Integrals - Complete Interactive Lesson

Part 1: Riemann Sums

∫ Definite Integrals

The Area Problem

Left, Right, and Midpoint Sums

Worked Example

Over- and Underestimates — Essential AP Skill

Trapezoidal Rule

Key Takeaways — Part 1

Part 2: Definite Integral Definition

∫ Definite Integrals

From Riemann Sums to Exact Area

Part 3: Properties of Integrals

∫ Definite Integrals

Essential Properties

Part 4: Fundamental Theorem of Calculus — Part 1

∫ Definite Integrals

The Fundamental Theorem — Part 1

Part 5: Fundamental Theorem of Calculus — Part 2

∫ Definite Integrals

The Evaluation Theorem (FTC Part 2)

Part 6: Mixed Integration Problems

∫ Definite Integrals

Combining All Tools

Strategy Guide

Part 7: Comprehensive Review

∫ Definite Integrals — Comprehensive Review

Complete Summary

Equal vs. Unequal Subintervals

Worked Example

Signed Area

Geometric Evaluation

Worked Example — Semicircle

Even and Odd Function Shortcuts

Worked Example

Key Takeaways — Part 2

Worked Examples — Given-Value Problems

Average Value of a Function

Worked Example

Key Takeaways — Part 3

With the Chain Rule

All Variations at a Glance

Worked Examples

Accumulation Functions

Connecting fff and FFF

Key Takeaways — Part 4

Quick Evaluation Examples

Net Change Theorem

Applications Table

Displacement vs Total Distance

Worked Example — Displacement vs Distance

Key Takeaways — Part 5

Absolute Value Integrals — Step by Step

Worked Example

Key Takeaways — Part 6

Common AP Exam Mistakes

Definite Integrals — Complete! ✅

AP Exam Checklist

Connecting $f$ and $F$