Evaluating Definite Integrals

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Textbook Reference

Opening Scenario

The area problem and the distance problem both lead to the same limit: a sum of many small products $f(x_i^*)\,\Delta x$ as $\Delta x \to 0$. This limit is so important that it gets its own name and notation: the definite integral.

But area and distance are always positive. What happens when $f$ dips below the $x$-axis? The definite integral is still defined there -- it just measures signed area, where regions below the $x$-axis contribute negatively.

Quick Reference

Definition. If $f$ is integrable on $[a, b]$, then $$\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x$$ where $\Delta x = \dfrac{b-a}{n}$ and $x_i^*$ is any point in the $i$-th subinterval.

Reading: “The integral from $a$ to $b$ of $f(x)$ with respect to $x$.”

Parts of the notation:

• $\int$ -- the elongated $S$ for “sum”
• $a, b$ -- the limits of integration (lower, upper)
• $f(x)$ -- the integrand
• $dx$ -- the variable of integration (analogous to $\Delta x$)

Geometric interpretation: $\displaystyle\int_a^b f(x)\,dx$ = (area above $x$-axis) $-$ (area below $x$-axis).

Key Concepts

1. The Definite Integral Is a Number

Unlike the derivative, which is a new function, the definite integral of $f$ on $[a, b]$ is a single real number. Once $a$, $b$, and $f$ are specified, $\int_a^b f(x)\,dx$ is determined. The variable $x$ inside is a dummy variable: $\int_a^b f(x)\,dx$, $\int_a^b f(t)\,dt$, and $\int_a^b f(u)\,du$ all denote the same number.

2. Signed Area

When $f(x) \geq 0$ on $[a, b]$, the definite integral equals the area of the region below $y = f(x)$ and above the $x$-axis -- precisely the area problem from Section 4.1.

When $f(x) < 0$ on part of $[a, b]$, the rectangles in that part have negative height $f(x_i^*)$, so they contribute negatively to the sum. The integral counts area above the $x$-axis as positive and area below the $x$-axis as negative.

Example. If $f(x) = \sin x$ on $[0, 2\pi]$, the first arch (above the axis on $[0, \pi]$) and the second arch (below the axis on $[\pi, 2\pi]$) have equal areas. They cancel, so $\displaystyle\int_0^{2\pi} \sin x\,dx = 0$.

3. Evaluating by Geometry

For simple functions, the definite integral can be read off from the geometry of the graph.

• Constant function: $\displaystyle\int_a^b c\,dx = c(b-a)$. The region is a rectangle.
• Linear function: $\displaystyle\int_a^b (mx + k)\,dx$ equals the signed area of a trapezoid.
• Semicircle: $\displaystyle\int_{-r}^{r} \sqrt{r^2 - x^2}\,dx = \dfrac{\pi r^2}{2}$. The region is a semicircle.

Geometric evaluation works only for simple shapes. For general functions, the Fundamental Theorem of Calculus (Section 4.3) provides the practical tool.

4. Evaluating by Definition

For polynomial functions, the summation formulas from Section 4.1 allow computation of the exact integral directly from the Riemann sum limit.

Worked Example

Evaluate $\displaystyle\int_0^2 (x - 1)\,dx$ using geometry.

The line $y = x - 1$ crosses the $x$-axis at $x = 1$.

On $[0, 1]$: triangle below the $x$-axis, base 1, height 1. Signed contribution: $-\dfrac{1}{2}$.

On $[1, 2]$: triangle above the $x$-axis, base 1, height 1. Signed contribution: $+\dfrac{1}{2}$.

$$\int_0^2 (x-1)\,dx = -\frac{1}{2} + \frac{1}{2} = 0.$$

Boxed answer: $\displaystyle\int_0^2 (x-1)\,dx = 0$.

Evaluate $\displaystyle\int_0^3 x^2\,dx$ using the definition.

$\Delta x = \dfrac{3}{n}$, $x_i = \dfrac{3i}{n}$.

$$\int_0^3 x^2\,dx = \lim_{n\to\infty}\frac{27}{n^3}\cdot\frac{n(n+1)(2n+1)}{6} = \lim_{n\to\infty}\frac{27(n+1)(2n+1)}{6n^2} = \frac{27 \cdot 2}{6} = 9.$$

Boxed answer: $\displaystyle\int_0^3 x^2\,dx = 9$.

Common Errors Summary

Leveled Practice

Level 1 -- Geometric Evaluation

Problem 1. Evaluate $\displaystyle\int_1^4 3\,dx$ using the geometry of the graph.

Level 2 -- Signed Area

Problem 2. Evaluate $\displaystyle\int_0^4 (2x - 4)\,dx$ using geometry.

Mastery Checklist

• State the definition of the definite integral as a limit of Riemann sums
• Identify the integrand, limits of integration, and variable of integration
• Evaluate a definite integral geometrically for constant and linear functions
• Explain the difference between signed area and total area
• Evaluate $\int_0^b x^2\,dx$ using the definition and summation formulas

Mental Model

The integral sign $\int$ is a stretched-out $S$ for “sum.” The $dx$ is the limiting form of $\Delta x$. The integrand $f(x)$ is the height. So $\int_a^b f(x)\,dx$ literally means “sum of $f(x)\,\Delta x$ as widths shrink to zero,” written in compact form.

The word “definite” signals that the limits $a$ and $b$ are fixed, so the result is a specific number, not a family of functions.

Connections

Looking back

• Riemann sums (Sections 4.1--4.2): The definite integral is the limit of these sums.

Looking ahead

• Properties of integrals (Section 4.2): Linearity and additivity simplify computation.
• Fundamental Theorem of Calculus (Section 4.3): Replaces the limit definition with antidifferentiation.

Back to Integration Foundations | Next: Properties of the Definite Integral