Introduction
 instanteneous rate of change
 slope of tangent
\( \frac{dy}{dx}[F(x)+c]=f(x)\)
 reverse process of derivative
If \( \frac{dy}{dx}F(x)=f(x)\) , then \( F(x)\) is called an antiderivative of \( f(x)\)  family of functions
the antidrivative of f(x) always infine in number, they are F(x)+c
For example, the antiderivative of \(3x^2\) is \(x^3+c\), it can be
\(x^3+1,x^3+2,x^3+3,x^3+4,x^3+\pi,x^3+\sqrt{2},x^3+e,x^3+0.3,\cdots\)
 a limiting process, defined as the limit of sum of rectangles under f(x)
 sum of products, adding peice of multiplication followed by addition: an infinite sum of extremely narrow rectangles, each rectangle having a height f(x) and a width dx; the differential of x, graphical area enclosed by the function and the interval boundaries
Given a function f(x), the antiderivative of f(x) is defined as a new function F(x) given by
F'(x)=f(x)
Blue curves are antiderivatives
Derivative and antiderivative
1. The derivative of f(x) is F(x): the slope of f(x) is represented by F(x)
2. The integral of F(x) is f(x): the mass of F(x) is represented by f(x)
Historical Evidence
 method of exhaustion in Greek (Eudoxus300BC)
 similar method in China (Liu Hui200AD)
 first proof of the fundamental theorem of calculus(Barrow1600AD)
 independent discovery of the fundamental theorem of calculus(Leibniz and Newton1600AD)
 notion were used [Lebniz∫, Fourierwith limits]
Integral as Riemann Sum
The integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. A tagged partition of a closed interval [a, b] on the real line is a finite sequence given by
\( a=x_0 \leq x_1 \leq x_2 \leq x_3 \leq ... \leq x_n=b\)
This partitions the interval [a, b] into n subintervals\( [x_{i−1}, x_i] \) of length \(\Delta x_i\) indexed by i.
A Riemann sum of a function f(x) with respect to such a tagged partition is defined as
\( \int_a^b f(x)dx= \displaystyle \sum _{i=1}^n f(x_i) \Delta x_i\)
where \(x_i\) can be any number inside the \( [x_{i−1}, x_i] \)
Lower Sum
Left Sum
Right Sum
Middle Sum
Graph of Riemann Sum
The Integral: Example
Use rectangles to estimate the area under the parabola \(y=0.5x^2+2x\) from 0 to 4.
Solution
The area is
Area=\(\displaystyle \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x\)
where
\(f(x_i)\) is the length of the rectangle
\(\Delta x\) is the breath of the rectangle
The length of the rectangle can be found from the
 upper point
 lower point
 left point
 right point
 middle point
Example
Evaluate \(\displaystyle \int_0^4(x^2)dx\)
Solution
With n subintervals, we have
\(\Delta x=\frac{ba}{n}=\frac{4}{n}\)
Using the right end points, we get
\(\displaystyle \int_0^4(x^2)dx\)  \( \displaystyle = \lim_{n \to \infty} \sum_{i=1}^n f \left ( \frac{4i}{n} \right) (\Delta x)\) 
\(\displaystyle= \lim_{n \to \infty} \sum_{i=1}^n \left ( \frac{4i}{n} \right)^2 \left ( \frac{4}{n} \right ) \) \(\displaystyle= \lim_{n \to \infty} \frac{64}{n^3} \sum_{i=1}^n (i^2) \) \(\displaystyle= \lim_{n \to \infty} \frac{64}{n^3} \frac{n(n+1)(2n+1)}{6} \) \(\displaystyle= \lim_{n \to \infty} 64 \frac{1(1+1/n)(2+1/n)}{6} \) \(= 64 \frac{1(1+0)(2+0)}{6} \) \( =\frac{64}{3} \) \( = 21.33\)  
Fundamental Theorem of Calculus
Mean Value theorem of Integral
Let f(x) : [a, b] → R be a continuous function. Then there exists c in [a, b] such that
\( \displaystyle \int _{a}^{b}f(x)dx=f(c)(ba)\)
Since the mean value of f(x) on [a, b] is defined as \( \frac{1}{ba} \int _a^b f(x)dx=f(c)\)
We can interpret the conclusion as f(x) achieves its mean value at some c in (a, b)
Thus
Let f(t) is continuous function on [x,x+h], then there exists c in [x,x+h] such that
\( \frac{1}{(x+h)x} \int_{x}^{x+h} f(t)dt=f(c)\)
or \( \frac{1}{h} \int_{x}^{x+h} f(t)dt=f(c)\)
Fundamental Theorem of Calculus,shows that
\( \frac{d}{dx} \left [ \int_a^b f(x)dx \right ]=f(x) \)
The fundamental theorem of calculus links the concept of differentiation and integration. It justifies the procedure by computing the integral using antiderivative
Thus, gives us a way to evaluate definite integrals without using Riemann sums or calculating areas.
Fundamental Theorem (First Part)
This theorem ensures that any integrable function has an antiderivative. Specifically, it guarantees the existence of antiderivative. It is also called theorem for Integrals and Antiderivatives
Statement
Let f(x) is continuous on [a, b] and F(x) is defined by
\( \displaystyle F(x)=\int _{a}^{x}f(t)dt\) for all x in [a, b]
Then
\( \displaystyle F'(x)=f(x)\) for all x in (a, b)
Proof
By definition, we choose a base point such that
\( \displaystyle F(x)=\int _a^x f(t)dt\)
(1)
\( \displaystyle F(x +h)=\int _{a}^{x +h}f(t)dt\)
(2)
Applying the definition of the derivative, we have
\(\displaystyle F'(x )=\displaystyle \lim_{h \to 0} \frac{F(x+h)F(x)}{h} \)
or \(\displaystyle F'(x )=\displaystyle \lim_{h \to 0} \frac{1}{h} \left [ F(x+h)F(x) \right ] \)
or \(\displaystyle F'(x )=\displaystyle \lim_{h \to 0} \frac{1}{h} \left [ \int _{a}^{x +h}f(t)dt \int _{a}^{x}f(t)dt \right ] \)Using (1) and (2)
or \(\displaystyle F'(x )=\displaystyle \lim_{h \to 0} \frac{1}{h} \left [ \int _{x}^{x +h}f(t)dt \right ] \)(3)
By Mean Value Theorem for Integrals in (3), there is some number c in [x,x+h] such that
\(\displaystyle F'(x )=\displaystyle \lim_{h \to 0} f(c) \)
Since c is between x and x + h, c approaches x as h approaches zero.
Thus
\(\displaystyle F'(x )=f(x) \)
This completes the proof.
Fundamental Theorem (Second Part)
The Fundamental Theorem of Calculus, Part 2 is also known as the evaluation theorem. It states that if we can find an antiderivative for the integrand, then we can evaluate the definite integral by evaluating the antiderivative at the endpoints of the interval and subtracting.
Statement
Let f(x) is integrable on [a, b], and f(x) admit its antiderivative F(x), then
\(\displaystyle F(b)F(a)=\int_{a}^{b}f(x)dx\)
Proof
Let \( g(x)=\int_a^x f(t)dt\), then by The Fundamental Theorem of Calculus Part 1, there exist g(x), which is an antiderivative of f(x), that is
\(g′(x)=f(x)\)
Also let, F(x) be antiderivative of f(x), then it follows that
\(F(x)=g(x)+c\) where C is a constant
Note: F(x) and g(x) differ only by a constant
Now
\( g(a)=\int_a^a f(t)dt=0\)
Next,
\(F(b)−F(a)=[g(b)+c]−[g(a)+c] \)
or \(F(b)−F(a)=g(b)−g(a) \)
or \(F(b)−F(a)=g(b)\)
or \(F(b)−F(a)=\int_a^b f(t)dt\)
or \( F(b)−F(a)=\int_a^b f(x)dx\)
Relation between Derivative, AntiDerivative and Integral
In the definite form, fundamental theorem of calculus connects differentiation with the definite integral as follows.
If f(x) is a continuous realvalued function defined on a closed interval [a, b], then integral of f over [a, b] is given by
\( \int_a^b\) f(x)dx=F(b)F(a)
For example,
मानौ, y=x एउटा function हो, जसको कुनै एउटा बिन्दुले Xaxis सँग बनाउने Area = \( \frac{1}{2}x^2\) छ।
यहाँ,
Integral of x is \( \frac{1}{2}x^2\) हुन्छ ।
Derivative of \( \frac{1}{2}x^2\) is x हुन्छ ।
त्यसैले, Integral is inverse process of differentiation भनिएको हो।
Derivative, AntiDerivative and Integral
Derivative, AntiDerivative and Integral are close due to Fundamental Theorem of Calculus (FTC). Before this theorem, antiderivative and integral were not thought of as related to each other.
Derivative and AntiDerivative
Derivative and Antiderivative are very close due to Fundamental Theorem of Calculus (FTC, I).
Derivative is an infinitesimal increment of change (difference), which is a quotient of differences(changes)
 a graphical slope (its “rise over run”),and slope of tangent line is the derivative of the function at that point
 specifid at a single point along the function where the slope is to be calculated
 a family of functions whose slopes are same
 The process of executing antiderivative is called antidifferentiation

AntiDerivative and Integral
AntiDerivative and Integral are very close due to Fundamental Theorem of Calculus (FTC, II).
Antiderivative of f(x) is F(x)+c such that [F(x)+c]'=f(x). a family of functions whose slopes are same
 The process of executing antiderivative is called antidifferentiation
 differential calculus, calculus of difference/change, is about how fast a dependent variable changes as independent variable changes
 sum of products, adding peice of multiplication followed by addition
 an infinite sum of extremely narrow rectangles, each rectangle having a height f(x) and a width dx; the differential of x, graphical area enclosed by the function and the interval boundaries
 The process of executing integral is called integration
 Integral calculus is the calculus of accumulation, is the sum of functions
 definite integral is \(\int _𝑎^𝑏 𝑥^2 𝑑𝑥=\frac{1}{3} (b^3 – a^3)=F(b) – F(a) \).
 indefinite integral is \(\int 𝑥^2 𝑑𝑥=\frac{x^3}{3}+c \) such that \(\frac{d}{dx} \left (\frac{x^3}{3} \right )=x^2\).

Derivative and Integral
Derivative and Integral are very close due to Fundamental Theorem of Calculus (FTC, I,II).
Derivative is an infinitesimal increment of change (difference), which is a quotient of differences (changes)
 a graphical slope (its “rise over run”) ,and slope of tangent line is the derivative of the function at that point
 specifid at a single point along the function where the slope is to be calculated
 differential calculus, calculus of difference/change, is about how fast a dependent variable changes as independent variable changes
 sum of products, adding peice of multiplication followed by addition
 an infinite sum of extremely narrow rectangles, each rectangle having a height f(x) and a width dx; the differential of x, graphical area enclosed by the function and the interval boundaries
 The process of executing integral is called integration
 Integral calculus is the calculus of accumulation, is the sum of functions
Type of Integral
 Integration as inverse operator
 Substitutio/Replacement: Derivative in numirator \( \int f^n.f' dx\) or \( \int \frac{f' dx}{f^n} \)
 By Evaluation
 By Parts
 Of the form: \([f(x)+f'(x)] e^x dx\)
 Partial Fraction
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