Definition of the Definite Integral

The definition of the definite integral is the foundation of integration theory; via Riemann sums it gives a rigorous meaning to “accumulation.”

Riemann sum definition

Definition of the definite integral

Let $f(x)$ be defined on $[a, b]$, partition it into $n$ subintervals:

$a = x_0 < x_1 < \cdots < x_n = b$

Pick any point $\xi_i$ in each $[x_{i-1}, x_i]$, and form

$S_n = \sum_{i=1}^n f(\xi_i) \Delta x_i$

where $\Delta x_i = x_i - x_{i-1}$. If $\lim_{\max\{\Delta x_i\}\to 0} S_n$ exists, it is the definite integral of $f(x)$ on $[a,b]$:

$\int_a^b f(x) \, dx = \lim_{\max\{\Delta x_i\} \to 0} \sum_{i=1}^n f(\xi_i) \Delta x_i$

$\xi_i$ (xi): Greek letter, pronounced “ksee”. Any point in subinterval $[x_{i-1}, x_i]$ (left/right/mid/any). Different choices give different Riemann sums, but as the mesh refines they converge to the same limit.

Definite integral formula

$\int_a^b f(x) \, dx = \lim_{\max\{\Delta x_i\} \to 0} \sum_{i=1}^n f(\xi_i) \Delta x_i$

$\int$ (integral sign): from the elongated “S” of Latin summa, introduced by Leibniz (1646–1716).
Reading: $\int_a^b f(x) \, dx$ — “integrate $f(x)$ from $a$ to $b$.”

Geometric meaning

$\int_a^b f(x) \, dx$ is the (signed) area of the curvilinear trapezoid bounded by $y=f(x)$, $x=a$, $x=b$, and the $x$-axis.

• If $f(x)\ge 0$, it’s the area.
• If $f(x)<0$, it’s the negative of the area.
• If $f(x)$ changes sign, it’s the algebraic sum of signed areas.

Existence

定理

If $f(x)$ is continuous on $[a,b]$, then $f(x)$ is integrable on $[a,b]$.

Corollaries:

1. Continuous functions on closed intervals are integrable.
2. Bounded functions with finitely many discontinuities are integrable.
3. Monotone functions on closed intervals are integrable.

练习题

练习 1

Using the Riemann-sum definition, approximate $\int_0^1 x \, dx$ with $n=4$ midpoints.

参考答案

思路：4 equal parts, $\Delta x=\tfrac14$, midpoints $\xi_i=\tfrac{1}{8},\tfrac{3}{8},\tfrac{5}{8},\tfrac{7}{8}$.
$S_4=\tfrac14(\tfrac18+\tfrac38+\tfrac58+\tfrac78)=\tfrac12$.
答案：$\tfrac12$。

练习 2

Explain the geometric meaning of $\int_{-1}^1 x \, dx$.

参考答案

思路：$f(x)=x$ is negative on $[-1,0]$, positive on $[0,1]$; areas $-\tfrac12$ and $+\tfrac12$, sum $0$.
答案：代数和为 0。

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\xi$	希腊字母	Xi（ksee）	子区间内任意取点
$\Delta x_i$	符号	Delta x_i	第 $i$ 子区间长度
$\sum$	希腊字母	Sigma	求和符号
$\lim$	数学符号	limit	极限
$\int$	数学符号	integral	定积分符号

中英对照

中文术语	英文术语	音标	说明
定积分	definite integral	/ˈdefɪnət ˈɪntɪɡrəl/	通过黎曼和极限定义的累积量
黎曼和	Riemann sum	/ˈriːmən sʌm/	分割区间后求和近似积分
子区间	subinterval	/sʌbˈɪntərvəl/	分割得到的区间
可积	integrable	/ˈɪntɪɡrəbəl/	定积分存在的性质
曲边梯形	curved trapezoid	/kɜːvd trəˈpiːzɔɪd/	由曲线围成的梯形区域
面积	area	/ˈeəriə/	图形的大小

NextProperties of the Definite Integral