∫ Riemann Sums

Part 1 of 7 — Riemann Sums

The Area Problem

How do we find the exact area under a curve? We approximate it using rectangles, then take the limit as the number of rectangles approaches infinity.

Left, Right, and Midpoint Sums

For $f(x)$ on $[a, b]$ with $n$ subintervals of width $\Delta x = \frac{b-a}{n}$:

$L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \quad \text{(Left Riemann Sum)}$

$R_n = \sum_{i=1}^{n} f(x_i) \Delta x \quad \text{(Right Riemann Sum)}$

$M_n = \sum_{i=1}^{n} f\left(\frac{x_{i-1}+x_i}{2}\right) \Delta x \quad \text{(Midpoint Sum)}$

Worked Example

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

$\Delta x = \frac{4-0}{4} = 1$. Left endpoints: $x = 0, 1, 2, 3$.

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

(The exact answer is $\frac{64}{3} \approx 21.33$, so $L_4 = 14$ is an underestimate since $f$ is increasing.)

Compute Riemann Sums 🎯

Trapezoidal Rule

The trapezoidal rule averages the left and right sums:

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

Worked Example

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

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

This is much closer to the exact value of $\frac{64}{3} \approx 21.33$.

AP Tip: The trapezoidal rule with data from a table is one of the most common AP free-response questions.

Trapezoidal Rule from a Table 🎯

Given the table:

Key Takeaways — Part 1

1. Riemann Sums approximate the area under a curve using rectangles
2. Left/Right/Midpoint use different sample points within each subinterval
3. For increasing functions: Left underestimates, Right overestimates
4. Trapezoidal Rule averages Left and Right sums for better accuracy
5. More rectangles ($n \to \infty$) means the sum approaches the exact integral

∫ The Definite Integral

Part 2 of 7 — Definite Integral Definition

From Riemann Sums to the Definite Integral

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

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

Geometric Interpretation

• $\int_a^b f(x)\,dx$ = signed area between $f(x)$ and the $x$-axis
• Area above the $x$-axis is positive
• Area below the $x$-axis is negative

Worked Example

Evaluate $\int_0^3 (2x + 1)\,dx$ geometrically.

This is a trapezoid with:

• Left height: $f(0) = 1$
• Right height: $f(3) = 7$
• Width: $3$

Area $= \frac{1}{2}(1 + 7)(3) = 12$

So $\int_0^3 (2x+1)\,dx = 12$, which we can verify: $\left[x^2 + x\right]_0^3 = (9+3) - 0 = 12$ ✓

Definite Integral Concepts 🎯

Important Properties

For odd functions ($f(-x) = -f(x)$) on symmetric intervals: $\int_{-a}^{a} f(x)\,dx = 0$

For even functions ($f(-x) = f(x)$) on symmetric intervals: $\int_{-a}^{a} f(x)\,dx = 2\int_0^a f(x)\,dx$

Evaluation via Antiderivatives

$\int_a^b f(x)\,dx = F(b) - F(a)$

where $F$ is any antiderivative of $f$.

Evaluate Definite Integrals 🎯

Key Takeaways — Part 2

1. The definite integral is the limit of Riemann sums
2. It represents signed area (above axis positive, below negative)
3. Odd functions integrate to 0 over symmetric intervals
4. Evaluate using the antiderivative: $\int_a^b f(x)\,dx = F(b) - F(a)$

∫ Properties of Integrals

Part 3 of 7 — Properties of Integrals

Essential Properties

Apply Integral Properties 🎯

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

Key Takeaways — Part 3

1. Integrals are linear: constants factor out, sums split
2. Additivity lets you break integrals into pieces over subintervals
3. Reversing limits flips the sign
4. These properties are essential for AP free-response questions with tables and given integral values

∫ Fundamental Theorem of Calculus — Part 1

Part 4 of 7 — FTC Part 1

The Big Idea

FTC Part 1 connects integration and differentiation as inverse operations:

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

In words: if you integrate a function and then differentiate, you get the original function back.

With Chain Rule

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

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

Worked Example 1

Find $\frac{d}{dx}\int_2^x t^3\,dt$

By FTC Part 1: the answer is simply $x^3$.

Worked Example 2

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

Upper limit is $g(x) = x^2$, so apply Chain Rule:

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

Worked Example 3

Find $\frac{d}{dx}\int_x^5 e^{t^2}\,dt$

Reverse limits first: $\int_x^5 = -\int_5^x$. Then:

$= -e^{x^2} \cdot 1 = -e^{x^2}$

FTC Part 1 🎯

Accumulation Functions

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

Key facts:

• $F(a) = 0$ (nothing accumulated yet)
• $F'(x) = f(x)$ (by FTC Part 1)
• $F$ is increasing where $f > 0$ and decreasing where $f < 0$
• $F$ has a local max where $f$ changes from positive to negative

AP Tip: FTC Part 1 with the Chain Rule is tested almost every year on the AP exam.

Accumulation Functions 🎯

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

Key Takeaways — Part 4

1. FTC Part 1: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$
2. With Chain Rule: $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$
3. Variable in lower limit: reverse limits first (adds a negative sign)
4. Accumulation functions connect the graph of $f$ to the behavior of $F$

∫ Fundamental Theorem of Calculus — Part 2

Part 5 of 7 — FTC Part 2 (Evaluation Theorem)

The Evaluation Theorem

$\int_a^b f(x)\,dx = F(b) - F(a)$

where $F$ is any antiderivative of $f$ (i.e., $F' = f$).

This is the practical computation tool: find an antiderivative, evaluate at the endpoints, subtract.

Notation

We write $F(x)\Big|_a^b$ or $\left[F(x)\right]_a^b$ to mean $F(b) - F(a)$.

Worked Examples

Evaluate Using FTC Part 2 🎯

Net Change Theorem

FTC Part 2 gives us the Net Change Theorem:

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

The integral of a rate of change gives the net change in the quantity.

Applications

Important: The integral of velocity gives displacement (net change), NOT total distance. For total distance, use $\int_a^b |v(t)|\,dt$.

Net Change Theorem 🎯

Key Takeaways — Part 5

1. FTC Part 2: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F' = f$
2. Net Change: the integral of a rate gives the net change in the quantity
3. Displacement vs Distance: $\int v\,dt$ = displacement; $\int |v|\,dt$ = total distance
4. Always check that your antiderivative is correct by mentally differentiating it

∫ Problem-Solving Workshop

Part 6 of 7 — Mixed Integration Problems

This workshop combines Riemann sums, FTC, properties of integrals, and applications.

AP-Style Mixed Problems 🎯

More Practice 🎯

Workshop Complete!

You combined all definite integral tools: Riemann sums, FTC, properties, symmetry, and applications.

∫ Review & Applications

Part 7 of 7 — Comprehensive Review

Complete Integration Summary

Final Assessment 🎯

Definite Integrals — Complete! ✅

You have mastered:

• ✅ Riemann Sums (left, right, midpoint, trapezoidal)
• ✅ Definite integral as a limit
• ✅ Properties of definite integrals
• ✅ FTC Part 1 (with and without Chain Rule)
• ✅ FTC Part 2 (evaluation)
• ✅ Net Change Theorem and applications