Riemann Sums, Summation Notation, and Definite Integral Notation
Introduced| English |
|---|
| summation notation/sʌˈmeɪʃn nəʊˈteɪʃn/ |
| definite integral/ˈdefɪnət ˈɪntɪɡrəl/ |
| limits of integration/ˈlɪmɪts ɒv ˌɪntɪˈɡreɪʃn/ |
Writing "add up all the rectangles" compactly
- A Riemann sum with many rectangles needs tidy notation — enter the summation notation 求和记号.
- $\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$ means "add $f(x_i)\,\Delta x$ for $i=1$ to $n$."
- Each strip has width $\Delta x=\dfrac{b-a}{n}$ and height $f(x_i)$ at a sample point.
- The $\Sigma$ just says "sum these rectangle areas."
For $n$ equal strips on $[a,b]$, the width $\Delta x$ is...
The interval width divided by the number of strips.
The symbol $\sum_{i=1}^{n}$ tells you to...
Sigma means sum over the index range.
Let the rectangles get infinitely thin
- The estimate improves as $n$ grows and $\Delta x$ shrinks. Take the limit $n\to\infty$.
- In that limit the rectangle sum becomes the exact area — no approximation left.
- That limiting value is the definite integral 定积分.
-
$$\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$$
The integral is the exact area
y = x²
As the rectangles thin to nothing, their sum becomes the exact shaded area — the definite integral.
The definite integral $\int_a^b f\,dx$ is the limit of a Riemann sum as...
Infinitely many infinitely-thin rectangles → exact area.
Reading the integral notation
- $\displaystyle\int_a^b f(x)\,dx$: the $\int$ is a stretched "S" for sum; $a$ and $b$ are the limits of integration 积分限.
- $f(x)$ is the integrand (what you're accumulating); $dx$ is the infinitely thin width.
- Read it "the integral of $f$ of $x$, from $a$ to $b$."
- It denotes the exact signed area between the curve and the $x$-axis over $[a,b]$.
In $\int_a^b f(x)\,dx$, the function $f(x)$ being accumulated is called the ____.
$a,b$ are the limits; $f(x)$ is the integrand.
It's still signed area
- Just like the Riemann sum, the definite integral is signed.
- Where $f>0$ the integral adds area; where $f<0$ it subtracts.
- So $\int_a^b f\,dx$ can be positive, negative, or zero.
- $\Delta x$ carries the direction: going from $a$ to $b$ with $a
A definite integral $\int_a^b f\,dx$ evaluates to a single number.
It is a signed-area number, not a function.
If $f(x)<0$ over all of $[a,b]$, then $\int_a^b f\,dx$ is...
Below-axis area is negative signed area.
The definite integral is a number (a signed area), not a function — don't confuse it with the indefinite integral (a family of antiderivatives, lesson 6.8). And $\int_a^b f\,dx$ is signed area: a curve dipping below the axis makes a negative contribution, so the integral is not the same as "total geometric area."
Express the area under $f(x)=x^2$ on $[0,2]$ as a definite integral.
- Width of each of $n$ strips: $\Delta x=\dfrac{2-0}{n}=\dfrac2n$.
- Riemann sum: $\displaystyle\sum_{i=1}^{n} f(x_i)\,\Delta x$.
- Exact area: $\displaystyle\int_0^2 x^2\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i)\,\Delta x$ (which equals $\tfrac83$).
A Riemann sum in summation notation is $\sum_{i=1}^{n} f(x_i)\,\Delta x$ with $\Delta x=\frac{b-a}{n}$. Its limit as $n\to\infty$ is the definite integral $\int_a^b f(x)\,dx$ — the exact signed area, with $a,b$ the limits of integration and $f(x)$ the integrand.