$Integral as Area infographic - Riemann Sums, the Fundamental Theorem, and Accumulation$

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Integral as Area

Riemann Sums, the Fundamental Theorem, and Accumulation

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The definite integral $\int_{a}^{b} f(x) \, dx$ measures the signed area between the curve $f(x)$ and the x-axis from $x = a$ to $x = b$ . The concept arises from Riemann sums: dividing the interval $[a, b]$ into $n$ subintervals, forming rectangles of width $\Delta x$ and height $f(x_i)$ , and summing all their areas. As $n \to \infty$ and $\Delta x \to 0$ , this sum converges to the exact integral. Regions where $f(x)$ is negative contribute negative area to the definite integral.

The Fundamental Theorem of Calculus (FTC) connects differentiation and integration. Part 1 states that if $F$ is the antiderivative of $f$ , then $\frac{d}{dx}\left[\int f(t) \, dt \text{ from } a \text{ to } x\right] = f(x)$ . Part 2 provides the evaluation shortcut: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ , where $F$ is any antiderivative of $f$ . Integration also computes displacement, net change, work, and probability - any quantity that accumulates continuously over a continuous variable. The indefinite integral $\int f(x) \, dx = F(x) + C$ finds families of antiderivatives, with the constant $C$ representing the entire family.

Key Facts

Definite integral = signed area under curve; negative where $f(x) < 0$
Riemann sum: $\sum f(x_i)\Delta x \to \int_{a}^{b} f(x) \, dx$ as $n \to \infty$
FTC Part 2: $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ , where $F'(x) = f(x)$
Power rule for integration: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ ( $n \neq -1$ )
$\int_{a}^{b} f(x) \, dx = -\int_{b}^{a} f(x) \, dx$ (reversing limits changes sign)
Net displacement ≠ total distance; total distance = ∫|v(t)| dt

Vocabulary

Definite integral: The limit of a Riemann sum giving the signed area under a curve $f(x)$ between $x = a$ and $x = b$ .
Antiderivative: A function $F(x)$ whose derivative is $f(x)$ ; also called an indefinite integral. Differs from another antiderivative by only a constant $C$ .
Riemann sum: An approximation of the area under a curve using rectangles; the basis for the definition of the definite integral.
Fundamental Theorem of Calculus: The theorem connecting differentiation and integration: differentiation and integration are inverse operations, and antiderivatives evaluate definite integrals.
Accumulation function: A function of the form $A(x) = \int_a^x f(t)\,dt$ that represents the accumulated area (or net change) from a fixed point $a$ up to $x$ .

Common Mistakes to Avoid

Forgetting the constant of integration for indefinite integrals. $\int f(x)\,dx = F(x) + C$ , not just $F(x)$ . The constant matters because it represents an entire family of antiderivatives.
Treating negative areas as zero. If $f(x)$ is negative on part of $[a,b]$ , the integral subtracts that region's area. The definite integral gives net signed area, not total area.
Applying the power rule to $\int \frac{1}{x}\,dx$ . $\int x^n\,dx = \frac{x^{n+1}}{n+1}$ fails when $n = -1$ (division by zero). Instead, $\int \frac{1}{x}\,dx = \ln|x| + C$ .
Confusing indefinite and definite integrals. $\int f(x)\,dx$ is a family of functions $+ C$ ; $\int_a^b f(x)\,dx$ is a single number (the area). The definite integral needs bounds.

Practice Questions

1 Evaluate ∫[0 to 3] (x² − 2x) dx. Identify any regions where the integrand is negative and how they affect the result.
2 Find the total area (not net area) between $f(x) = x^2 - 4$ and the $x$ -axis from $x = 0$ to $x = 3$ .
3 A particle moves with velocity $v(t) = t^2 - 4t + 3$ m/s. Find the displacement and total distance traveled from $t = 0$ to $t = 4$ s.

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