Chapter 2: Definite Integral
Definite Integral
We study the definition of the definite integral and the Fundamental Theorem of Calculus. We understand its relationship with area.
2.1 The Area Problem
How do we find the area of the region enclosed by the curve $y = f(x)$, the $x$-axis, and the vertical lines $x = a$ and $x = b$?
Riemann Sum Method
- Divide the interval $[a, b]$ into $n$ equal subintervals
- Compute the area of the rectangle on each subinterval
- Sum the areas of all rectangles
- Take the limit as $n \to \infty$
Dividing $[a, b]$ into $n$ equal parts, the partition points are:
$$x_k = a + \frac{b-a}{n}k \quad (k = 0, 1, 2, \ldots, n)$$The width of each subinterval is $\Delta x = \dfrac{b-a}{n}$
The sum of the rectangle areas over each subinterval $[x_{k-1}, x_k]$ (Riemann sum):
$$S_n = \sum_{k=1}^{n} f(x_k) \cdot \Delta x = \sum_{k=1}^{n} f(x_k) \cdot \frac{b-a}{n}$$2.2 Definition of the Definite Integral
Definition: Definite Integral
The definite integral of a function $f(x)$ over the interval $[a, b]$ is defined as:
$$\int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \cdot \frac{b-a}{n}$$When this limit exists, we say $f(x)$ is integrable on $[a, b]$.
Terminology:
- $a$: lower limit of integration
- $b$: upper limit of integration
- $\displaystyle\int_a^b$: read as "the integral from $a$ to $b$"
Definite vs. Indefinite Integral
- Indefinite integral $\displaystyle\int f(x)\,dx$: a function (family of antiderivatives)
- Definite integral $\displaystyle\int_a^b f(x)\,dx$: a number
2.3 Fundamental Theorem of Calculus
Theorem: Fundamental Theorem of Calculus (First Form)
If $f(x)$ is continuous on $[a, b]$ and $F(x)$ is an antiderivative of $f(x)$ (i.e., $F'(x) = f(x)$), then:
$$\int_a^b f(x) \, dx = F(b) - F(a)$$Notation: We write $F(b) - F(a)$ as $[F(x)]_a^b$ or $\left.F(x)\right|_a^b$.
Thanks to this theorem, computing a definite integral reduces to:
- Finding an antiderivative $F(x)$ of $f(x)$
- Evaluating $F(b) - F(a)$
There is no need to compute the limit of Riemann sums directly.
Example 2.1
Evaluate $\displaystyle\int_0^2 x^2 \, dx$.
An antiderivative of $x^2$ is $\dfrac{x^3}{3}$.
$$\int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{2^3}{3} - \frac{0^3}{3} = \frac{8}{3} - 0 = \frac{8}{3}$$Theorem: Fundamental Theorem of Calculus (Second Form)
If $f(x)$ is continuous, then for the function $G(x) = \displaystyle\int_a^x f(t) \, dt$:
$$G'(x) = f(x)$$This means that differentiating a definite integral with respect to its upper limit recovers the integrand.
2.4 Properties of Definite Integrals
Property 1: Linearity
$$\int_a^b \{cf(x) + dg(x)\} \, dx = c\int_a^b f(x) \, dx + d\int_a^b g(x) \, dx$$Property 2: Interval Splitting
If $a < c < b$:
$$\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$$Property 3: Reversing Limits
$$\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$$Property 4: Equal Limits
$$\int_a^a f(x) \, dx = 0$$Property 5: Inequality
If $f(x) \leq g(x)$ on $[a, b]$, then:
$$\int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx$$Property 6: Even and Odd Functions
If $f(x)$ is an even function ($f(-x) = f(x)$):
$$\int_{-a}^{a} f(x) \, dx = 2\int_0^a f(x) \, dx$$If $f(x)$ is an odd function ($f(-x) = -f(x)$):
$$\int_{-a}^{a} f(x) \, dx = 0$$2.5 Worked Examples
Example 2.2
Evaluate $\displaystyle\int_1^3 (2x + 1) \, dx$.
Example 2.3
Evaluate $\displaystyle\int_0^{\pi} \sin x \, dx$.
Example 2.4: Using Even Functions
Evaluate $\displaystyle\int_{-2}^{2} x^4 \, dx$.
Since $f(x) = x^4$ is an even function:
$$\int_{-2}^{2} x^4 \, dx = 2\int_0^2 x^4 \, dx = 2\left[\frac{x^5}{5}\right]_0^2 = 2 \cdot \frac{32}{5} = \frac{64}{5}$$Example 2.5: Using Odd Functions
Evaluate $\displaystyle\int_{-1}^{1} x^3 \, dx$.
Since $f(x) = x^3$ is an odd function:
$$\int_{-1}^{1} x^3 \, dx = 0$$Verification:
$$\int_{-1}^{1} x^3 \, dx = \left[\frac{x^4}{4}\right]_{-1}^{1} = \frac{1}{4} - \frac{1}{4} = 0 \quad \checkmark$$Example 2.6: Interval Splitting
Evaluate $\displaystyle\int_0^2 |x - 1| \, dx$.
The sign of $|x - 1|$ changes at $x = 1$:
$$|x - 1| = \begin{cases} 1 - x & (0 \leq x < 1) \\ x - 1 & (1 \leq x \leq 2) \end{cases}$$Splitting the interval:
$$\begin{align} \int_0^2 |x - 1| \, dx &= \int_0^1 (1 - x) \, dx + \int_1^2 (x - 1) \, dx \\ &= \left[x - \frac{x^2}{2}\right]_0^1 + \left[\frac{x^2}{2} - x\right]_1^2 \\ &= \left(1 - \frac{1}{2}\right) - 0 + \left(2 - 2\right) - \left(\frac{1}{2} - 1\right) \\ &= \frac{1}{2} + 0 + \frac{1}{2} = 1 \end{align}$$Checklist for Computing Definite Integrals
| Check | Approach |
|---|---|
| Does the integrand contain absolute values? | Split the interval where the sign changes |
| Is the integrand even or odd? | Use the formula for symmetric intervals |
| Order of the limits | If $a > b$, watch the sign |
2.6 Exercises
Problem 1
Evaluate the following definite integrals.
- $\displaystyle\int_0^1 (3x^2 - 2x + 1) \, dx$
- $\displaystyle\int_1^4 \sqrt{x} \, dx$
- $\displaystyle\int_0^{\pi/2} \cos x \, dx$
- $\displaystyle\int_1^e \frac{1}{x} \, dx$
Show Solution
- $$\left[x^3 - x^2 + x\right]_0^1 = (1 - 1 + 1) - 0 = 1$$
- $$\left[\frac{2}{3}x^{3/2}\right]_1^4 = \frac{2}{3}(8 - 1) = \frac{14}{3}$$
- $$\left[\sin x\right]_0^{\pi/2} = 1 - 0 = 1$$
- $$\left[\ln|x|\right]_1^e = \ln e - \ln 1 = 1 - 0 = 1$$
Problem 2
Evaluate the following definite integrals.
- $\displaystyle\int_{-3}^{3} x^5 \, dx$
- $\displaystyle\int_{-1}^{1} (x^2 + x^3) \, dx$
- $\displaystyle\int_0^3 |2x - 4| \, dx$
Show Solution
-
Since $x^5$ is an odd function, $\displaystyle\int_{-3}^{3} x^5 \, dx = 0$
-
$x^2$ is even and $x^3$ is odd, so:
$$\int_{-1}^{1} (x^2 + x^3) \, dx = \int_{-1}^{1} x^2 \, dx + 0 = 2\int_0^1 x^2 \, dx = 2 \cdot \frac{1}{3} = \frac{2}{3}$$ -
Since $2x - 4 = 0$ at $x = 2$, the sign changes there:
$$\begin{align} &= \int_0^2 (4 - 2x) \, dx + \int_2^3 (2x - 4) \, dx \\ &= \left[4x - x^2\right]_0^2 + \left[x^2 - 4x\right]_2^3 \\ &= (8 - 4) - 0 + (9 - 12) - (4 - 8) \\ &= 4 + (-3) - (-4) = 5 \end{align}$$
Problem 3
Evaluate $\displaystyle\frac{d}{dx}\int_0^x t^2 e^t \, dt$.
Show Solution
By the Fundamental Theorem of Calculus (Second Form):
$$\frac{d}{dx}\int_0^x t^2 e^t \, dt = x^2 e^x$$