Integral

---

---

---

Introduction

Solution 👉 Click Here

Derivative

1. instanteneous rate of change
2. slope of tangent

Anti-Derivative Let f(x) be a continuous function defined in an open interval (a,b), then a function F(x)+c is called an antiderivative of f(x) if
\( \frac{dy}{dx}[F(x)+c]=f(x)\)

1. reverse process of derivative
   If \( \frac{dy}{dx}F(x)=f(x)\) , then \( F(x)\) is called an anti-derivative of \( f(x)\)
2. family of functions
   the anti-drivative of f(x) always infine in number, they are F(x)+c
   For example, the anti-derivative of \(3x^2\) is \(x^3+c\), it can be
   \(x^3+1,x^3+2,x^3+3,x^3+4,x^3+\pi,x^3+\sqrt{2},x^3+e,x^3+0.3,\cdots\)

Integral

1. a limiting process, defined as the limit of sum of rectangles under f(x)
2. 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

Mathematically

Given a function f(x), the anti-derivative of f(x) is defined as a new function F(x) given by
F'(x)=f(x)

Blue curves are anti-derivatives

Blue curves are anti-derivatives

Derivative and anti-derivative
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

1. method of exhaustion in Greek (Eudoxus-300BC)
2. similar method in China (Liu Hui-200AD)
3. first proof of the fundamental theorem of calculus(Barrow-1600AD)
4. independent discovery of the fundamental theorem of calculus(Leibniz and Newton-1600AD)
5. notion were used [Lebniz-∫, Fourier-with limits]

---

---

---

Integral as Riemann Sum

Solution 👉 Click Here

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 sub-intervals\( [x_{i−1}, x_i] \) 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 \( [x_{i−1}, x_i] \)

Upper Sum

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

1. upper point
2. lower point
3. left point
4. right point
5. middle point

Example

Evaluate \(\displaystyle \int_0^4(x^2)dx\)
Solution
With n subintervals, we have
\(\Delta x=\frac{b-a}{n}=\frac{4}{n}\)
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

Solution 👉 Click Here

Before, we start Fundamental Theorem of Calculus(FTC), we will prove Mean Value theorem of Integral.

Mean Value theorem of Integral

Let f(x) : [a, b] → R be a continuous function. Then there exists c in [a, b] such that
\( \displaystyle \int _{a}^{b}f(x)dx=f(c)(b-a)\)
Since the mean value of f(x) on [a, b] is defined as \( \frac{1}{b-a} \int _a^b f(x)dx=f(c)\)
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 \( \frac{1}{h} \int_{x}^{x+h} f(t)dt=f(c)\)

Now
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, Anti-Derivative and Integral

Solution 👉 Click Here

In the definite form, fundamental theorem of calculus connects differentiation with the definite integral as follows.
If f(x) is a continuous real-valued 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 X-axis 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, Anti-Derivative and Integral

Derivative, Anti-Derivative and Integral are close due to Fundamental Theorem of Calculus (FTC). Before this theorem, anti-derivative and integral were not thought of as related to each other.

1. Derivative and Anti-Derivative

   Derivative and Anti-derivative 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
   Anti-derivative 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
   Both are part of differential calculus, calculus of difference/change, is about how fast a dependent variable changes as independent variable changes

2. Anti-Derivative and Integral

   Anti-Derivative and Integral are very close due to Fundamental Theorem of Calculus (FTC, II).
   Anti-derivative 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
   Integral of f(x) is defined by 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
   • The process of executing integral is called integration
   • Integral calculus is the calculus of accumulation, is the sum of functions
   There is a very small difference between integral f(x) and anti-derivative F(x), let \(f(x) = x^2\), then,
   1. definite integral is \(\int _𝑎^𝑏 𝑥^2 𝑑𝑥=\frac{1}{3} (b^3 – a^3)=F(b) – F(a) \).
   2. indefinite integral is \(\int 𝑥^2 𝑑𝑥=\frac{x^3}{3}+c \) such that \(\frac{d}{dx} \left (\frac{x^3}{3} \right )=x^2\).

3. 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
   Integral of f(x) is defined by 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
   • The process of executing integral is called integration
   • Integral calculus is the calculus of accumulation, is the sum of functions
   Just as division and multiplication are inverse mathematical functions, differentiation and integration are also inverse mathematical functions. differentiation always divides while integration always multiplies.

Type of Integral

1. Integration as inverse operator
2. Substitutio/Replacement: Derivative in numirator \( \int f^n.f' dx\) or \( \int \frac{f' dx}{f^n} \)
3. By Evaluation
4. By Parts
5. Of the form: \([f(x)+f'(x)] e^x dx\)
6. Partial Fraction