Introduction
- instanteneous rate of change
- slope of tangent
Let f(x) be a continuous function defined in (a,b), then F(x)+c is called an antiderivative of f(x) if
\( \frac{dy}{dx}[F(x)+c]=f(x)\)
- reverse process of derivative
If \( \frac{dy}{dx}F(x)=f(x)\) , then \( F(x)\) is called an anti-derivative of \( f(x)\) - 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
\(x^3+1,x^3+2,x^3+3,x^3+4,x^3+\pi,x^3+\sqrt{2},x^3+e,x^3+0.3,\cdots\)
- a limiting process, defined as the limit of sum of rectangles under f(x)
- sum of products: an infinite sum of narrow rectangles, having a height f(x) and a width dx
Given a function f(x), the anti-derivative of f(x) is defined by F(x) s. t.
F'(x)=f(x)
Blue curves are anti-derivatives
Derivative and Integral
- The derivative of f(x) is F(x): the slope of f(x) is represented by F(x)
- 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 (Eudoxus-300BC)
- similar method in China (Liu Hui-200AD)
- first proof of the fundamental theorem of calculus(Barrow-1600AD)
- independent discovery of the fundamental theorem of calculus(Leibniz and Newton-1600AD)
- notion were used [Lebniz-∫, Fourier-with 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, b] 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 [a, b] into n sub-intervals\( [x_{i−1}, x_i] \) of length \(\Delta x_i\) indexed by \(i\).
A Riemann sum of 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 \( [x_{i−1}, x_i] \) based on different approaches.
These approaches are as below.
Upper sum
Lower sum
Left sum
Right sum
Middle sum
Graph of Riemann Sum
The Integral
The integral 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
\(\Delta x=\frac{b-a}{n}=\frac{4}{n}\)
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
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
\( \displaystyle \int _{a}^{b}f(x)dx=f(c)(b-a)\)
Thus
Let f(t) is continuous function on [x,x+h], then there exists c in [x,x+h] such that
\( \int_{x}^{x+h} f(t)dt=h f(c)\)
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 write
\( \displaystyle F(x)=\int _a^x f(t)dt\) (1)
\( \displaystyle F(x +h)=\int _{a}^{x +h}f(t)dt\)
(2)
Subtracting (10 from (2), we get
\( F(x+h)-F(x)=\int _{a}^{x +h}f(t)dt-\int _{a}^{x}f(t)dt\)
or\( F(x+h)-F(x)=\int _{a}^{x +h}f(t)dt+\int _{x}^{a}f(t)dt\)
or\( F(x+h)-F(x)=\int _{x}^{a}f(t)dt+\int _{a}^{x +h}f(t)dt \)
or\( F(x+h)-F(x)=\int _{x}^{x+h}f(t)dt \)
Applying mean value theorem in right part, we get
\( F(x+h)-F(x)=h f(c)\)c in [x, x+h]
or\( \dfrac{F(x+h)-F(x)}{h} =f(c)\)c in [x, x+h]
Taking limit as \(h \to 0\), we get
\( \displaystyle \lim_{h \to 0} \dfrac{F(x+h)-F(x)}{h} =\lim_{h \to 0} f(c)\)
or\( F'(x) =f(x)\)
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
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
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.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
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
Integral
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 narrow rectangles, each rectangle having a height f(x) and a width dx
- 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 \(\frac{d}{dx} \left (\frac{x^3}{3} \right )=x^2\).
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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