Definite Integrals

∫ Definite Integrals

Part 1 of 7 — Riemann Sums

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

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 underestimate.

Over- and Underestimates — Essential AP Skill

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

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).

Riemann Sum Computations 🎯

Trapezoidal Rule

The Trapezoidal Rule averages the Left and Right sums:

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

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:

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

Worked Example

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!

Trapezoidal Rule from a Table 🎯

Given the table:

Classify each approximation. 🔍

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

Compute a Riemann Sum. ✍️

Key Takeaways — Part 1

Up Next: Part 2 — The Definite Integral.

∫ 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 $n \to \infty$:

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

Signed Area

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

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:

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 $r = 3$ = $\frac{9\pi}{2}$

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

Definite Integral Concepts 🎯

Even and Odd Function Shortcuts

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

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

Up Next: Part 3 — Properties of Integrals.

∫ Definite Integrals

Part 3 of 7 — Properties of Integrals

Essential Properties

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

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

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

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 $f_{\text{avg}}$ 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 $f$ is continuous). They may ask you to find this $c$.

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

Up Next: Part 4 — FTC Part 1.

∫ Definite Integrals

Part 4 of 7 — FTC Part 1

The Fundamental Theorem — Part 1

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

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

With the Chain Rule

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

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

$g(x) = x^2$, $g'(x) = 2x$: $\sin(x^2) \cdot 2x = 2x\sin(x^2)$

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

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

FTC Part 1 🎯

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 $F$

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

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

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

∫ Definite Integrals

Part 5 of 7 — FTC Part 2 & Net Change

The Evaluation Theorem (FTC Part 2)

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

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

Quick Evaluation Examples

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

Displacement vs Total Distance

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

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

Up Next: Part 6 — Problem-Solving Workshop.

∫ 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

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

1. Find where $f(x) = 0$ (the zeros)
2. Determine sign of $f$ on each subinterval
3. Split the integral at each zero
4. 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 $|x-2| = -(x-2) = 2-x$
• On $[2,4]$: $x - 2 > 0$, so $|x-2| = x-2$

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

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

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 ✍️

Key Takeaways — Part 6

Up Next: Part 7 — Comprehensive Review.

∫ Definite Integrals — Comprehensive Review

Part 7 of 7 — Final Assessment

Complete Summary

Common AP Exam Mistakes

Final Assessment — Set 1 🎯

Final Assessment — Set 2 🎯

AP-Style Comprehensive 🔍

Final Challenge ✍️

Definite Integrals — Complete! ✅

You have mastered:

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 $f$

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