Integral, integration

• Integral

 

 

1) While changing x from a to b

2) Multipliy f(x) by dx

3) Sum all of 2)

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• dx = ∆x = the amount of change in x
• x : change from x1 --> to x2 then, ∆x = x2-x1

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Indefinite Integral vs Definite Integral

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• Definite Integral

 

 

A NUMBER and represents the area under the curve f(x) from x=a to x=b.

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For example,

 

Q. There is a function(Definite Integral) log x, how can get the all the area of log x(from 0 to x)?

 

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log x : A function that has been set to make this area x.

The graph : If you integrate the area of x, is equal to the length of x.

 

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- Thinking

Let's calculate the area of rectangle. Then it will be easy to substract log x area from the rectangle.

 

x-axis : x

y-axis : log x

area of rectangle : x * log x

 

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- Answer

 

(x * log x) - x + C = ∫ log x * dx

 

Area of Definite Integral log x : (x * log x) - x + C

C : a pre-given thing that has not changed, a starting point

 

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Mathematical representation of Area of Definite Integral log x : ∫ log x * dx

 

∫ : Sum all the rectangles

height : log x

x : take out the moment when x clicks while x changes continuously Iinfinitely small moment).

d :  Sum x

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d : Differential

 

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• Indefinite Integral

 

 

A FUNCTION and answers the question, "What function when differentiated gives f(x)?"

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• Decomposition of Integral

 

1. ∫

 

Integral, meaning Sum.

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2. f(x)

 

y = f(x)

 

The value of y, the red line in the figure above.

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3. dx

 

The instantaneous change of x.

 

For example,

 

△x : The base line

△x/2 : Cut it vertically and divide it into two.

limδx→0 : Continue to divide this rectangle indefinitely, the base line will be close to 0.

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3-1. △x

 

The specific interval change of x.

If x changed from x1 to x2, then △x = x2-x1.

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4.  f(x)dx

 

 

While changing x from a to b

and f(x) * dx-->sum of them-->Σf(x) * dx

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f(x)dx : The area of red bars.

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For example,

 

 

- f(x) : radius

 

- the area of a circle :$\pi f(x{)}^2$

 

∴ the volume of a disc (cylindrical) : $V={\int }_a^b \pi f(x{)}^{2}dx$

 

https://www.youtube.com/watch?v=FjPem4Jzses&list=PLBPUzs26iQhpxyBcHSs-dI9pW19jaslcq

https://darkpgmr.tistory.com/45 

https://www.youtube.com/watch?v=FjPem4Jzses&list=PLBPUzs26iQhpxyBcHSs-dI9pW19jaslcq&index=1

https://www.youtube.com/watch?v=t99QiDaFs1c&list=PLBPUzs26iQhpxyBcHSs-dI9pW19jaslcq&index=4