Lecture 31: Basic Concepts of Definite Integrals

Common Question Types and Typical Examples in This Chapter

Exam Content

(1) Concept of definite integrals
(2) Properties of definite integrals
(3) Functions of the upper limit of integrals
(4) Calculation of definite integrals

Frequently Tested Question Types

Question Type 1: Concept, properties, and geometric meaning of definite integrals
Question Type 2: Calculation of Definite Integrals (Key Point)
Question Type 3: Upper Limit Points (Key Point)

1.1 Basic Concepts of Definite Integrals

1.1.1 Definition of Definite Integrals

#####Definition: Definiteintegral

description: $f(x)$ is bounded on $[a,b]$, arbitrarily inserting a node on $[a,b]$ to divide into n intervals $\Delta x_{1}\Delta x_{2}\cdots\Delta x_{n}$, and any point i is chosen, we have: $$\int_{a}^{b}f(x)dx=\lim_{λ\to0}\sum_{x=1}^{n}f(\xi_{i})\Delta x_{i}$$

Where: $$\lambda=\max\{\Delta x_{1}\cdots\Delta x_{n}\}$$

Explanation

Concept:
A definite integral is the**sum limit of $f(x)$ on $[a,b]$;
1. Divide: divide the $[a,b]$ interval into several segments;
Let function $f(x)$ be continuous on interval $[a, b]$, divide interval $[a, b]$ into n subintervals $[x_0,x_1]$, ($x_1,x_2],(x_2,x_3],\ldots,(x_{n-1},x_n]$, where $x_0 = a$ and $x_n = b$
2. Even: to take a section of $[x_{i-1},x_i]$ from it;
3. Product: sum all subsegments: $\sum_{x=1}^{n}f(\xi_{i})\Delta x_{i}$
4. Precision: Narrowing each segment toward infinitesimal: $\lim_{λ\to0}\sum_{x=1}^{n}f(\xi_{i})\Delta x_{i}$
Summary:
Definite integrals are a special limit;
The existence problem of definite integrals -> The integrability problem of limits;
Segments:
$\lambda=\max\{\Delta x_1,\Delta x_2,\ldots,\Delta x_n\}$, where $\lambda$ is the maximum subinterval length.
If there exists a limit of the sum of integrals when λ→0, then this limit is called the definite integral of the function f(x) on the interval $[a, b]$, denoted as $\int_a^bf(x)dx$, and is said to be integrable on the interval $[a, b]$;
Note:
1. $\lambda\to0$ is not equivalent to $n\to\infty$.
2. $\int_a^bf(x)\,dx$ depends only on $f$ and the interval $[a,b]$.
3. The limit $\lim_{\lambda\to0}\sum_{i=1}^nf(\xi_i)\Delta x_i$ is independent of the choice of sample points $\xi_i$ and the partition of $[a,b]$.
Because it is unrelated to the method of division, and since equal division is the simplest, equal division is used;
$$\int_{0}^{1}f(x)\operatorname{d}x=\lim_{\lambda\to0}\sum_{i=1}^{n}f(\xi_i)\Delta x_{i}=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}f(\frac{i}{n})$$
This allows us to find the limit of a sum to be the definite integral of it;
Here $\frac{1}{n}$ is the common subinterval width.
4. Definite integrals yield a specific number, which depends only on the integrand and the integration interval, and is independent of the integral variable;

1.1.2 Conditions for the existence of definite integrals

#####Theorem: Definiteintegralexistence

description:

1. Condition 1: $f(x)$ is continuously -> integrable on $[a,b]$;

2. Condition 2: $f(x)$ is bounded on $[a,b]$ and has a finite number of discontinuities -> that are integrable;

3. Condition 3: $f(x)$ There are only a finitely many Type I discontinuities on $[a,b]$;

Explanation

Among them, 1 and 3 are used more frequently;
Requirements:
Integrable -> bounded;
But bounded cannot derive a integrable existence;
Summary:
Being bounded is a necessary condition for the existence of definite integrals;
An integrable integral is a sufficient condition for the existence of definite integrals;

1.1.3 Geometric Meaning of Definite Integrals

Function values can be positive or negative

Concept:
The definite integral equals the area of the graph of the function on geometry;
When the function value is positive or negative, the current definite integral equals the sum of the areas of parts A and C, minus the area size of part B;
The sign of the integral must be determined from the function and the interval.
Illustration:

1.2 Properties of Definite Integrals

1.2.1 Basic Properties: Inequalities

Property 1: Inequality property

$\text{ If }f(x)\leq g(x),\text{ then }\int_a^bf(x)\operatorname{d}x\leq\int_a^bg(x)\operatorname{d}x$
Note:
a needs to be less than or equal to b

Nature 2: Valuation

$m(b-a)\leq\int_a^bf(x)\operatorname{d}x\leq M(b-a)$
Definite integral between the maximum and minimum values;

Property 3: Absolute value

$\left|\int_a^bf(x)\mathbf{d}x\right|\leq\int_a^b\lvert f(x)\rvert\mathbf{d}x.$

1.2.2 Basic Properties: Mean Value Theorem

Nature 1

If $f(x)$ is continuous on $[a,b]$, then
$$\int_a^bf(x)\operatorname{d}x=f(\xi)(b-a),a<\xi<b$$

Nature 2

$g(x)$ does not change sign if $f(x), g(x)$ continuity on $\left[a, b\right]$
$$\int_a^bf(x)g(x)\operatorname{d}x=f(\xi)\int_a^bg(x)\operatorname{d}x,a<=\xi<=b$$

Applications

1. Proof problems for integrals;
2. Find the limit related to the integral;

1.2.2 Other Properties

Two Provisions

1. If b = a, the definite integral from a to a equals 0;
2. The definite integral from a to b equals the inverse of the definite integral from b to a;
$\int_{a}^{b}f\left(x\right)dx=-\int_{b}^{a}f\left(x\right)dx$
Exchange upper and lower limits;

Nature 1:$\int_{a}^{b}(\alpha f(x)+\beta g(x))dx=\alpha\int_{a}^{b}f(x)dx+\beta\int_{a}^{b}g(x)dx$

Nature 2:

If $a<c<b$ (c is inside)
Then $\int_{a}^{b}f\left(x\right)dx=\int_{a}^{c}f\left(x\right)dx+\int_{c}^{b}f\left(x\right)dx$
If $a<b<c$ (c is outside)
Then $\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx-\int_{b}^{c}f(x)dx$ $=\int_{a}^{c}f\left(x\right)dx+\int_{c}^{b}f\left(x\right)dx$
So, the final conclusion: $\int_{a}^{b}f\left(x\right)dx=\int_{a}^{c}f\left(x\right)dx+\int_{c}^{b}f\left(x\right)dx$

Nature 3:$\begin{aligned}f(x)\equiv1;\,\int_{a}^{b}1dx=b-a\\\int_{a}^{b}kdx=k(b-a)\end{aligned}$

Property 4:M and m are the maximum and minimum values respectively, so $m(b-a)\leq\int_{a}^{b}f(a)dx\leq M(cb-a)$

1.3 Frequently Tested Question Types

Question Type: Theconceptofdefiniteintegrals. Properties and Geometric Meaning

PART 1: Problem-solving methods

Question Type: n Items and Addition

Approach:
1. Use the squeeze theorem;
2. Defining using definite integrals;
Solving definite integrals:
Step 1: First, present the cute silver $1/n$
Requirement: The element after the proposal is $1/n$ and cannot be offset by unlikable factors;
Because cute silver is the $\Delta x_{i}$ in $\int_{a}^{b}f(x)dx=\lim_{λ\to0}\sum_{x=1}^{n}f(\xi_{i})\Delta x_{i}$
Step 2: Observe the formula after being proposed, find the variable -> Wherever it changes, that is the variable;
Step 3: Obtain the interval -> from where it changes to -> Find the upper and lower bounds of the definite integral;
How to judge:
Generally, the squeeze theorem is used first, and if it is not available, definite integrals are used to define it;

Question Type: Geometric Meaning of Definite Integrals

Note:
When the upper and lower limits are required to be negative, such as the value of the definite integral in the $(-3,0)$ part, it is important to note: the upper limit of the definite integral cannot be less than the lower bound;
For example, $F(-2)=\int_{0}^{-2}f(t)dt$ This formula is incorrect because the upper limit cannot be less than the lower limit. This form must be used: $F(-2)=-\int_{-2}^{0}f(t)dt$
Illustration: