Author Archives: peiyao zhang

The difference between integration and antidifferentiation

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Definition

Antidifferentiation:

If f(x) is a function, then an antiderivative of f(x) is another function F(x), such that F'(x)=f(x).

Intergral:

If f(x) is a function, then the integral of f(x) over the interval [a,b] is the limit of its Riemann Sum over that interval (if it converges).

Difference

For anti differentiation , if there is an anti derivative, there are infinite function F(x): F(x)+1, F(x)+1/2, F(x)+pi, they can be concluded by F(x)+r,r is any real number. There is no special antiderivative for a function, they’re all on equal footing.

However, integral gives the area under the function f(x) between x=a and x=b.

This means that the integral of a function is a number that is special.

Take an example, we have a function f(x)=3x^2.

The anti derivative can be F(x)=x^3+1,or F(x)=x^3+3…

However,the integral from 1 to 2 is \int_1^2 3x^2 dx=8, which is special.

So we know that anti derivative is not special for a function but integral is special for a function.

Denises Wang 24482168

Yuki Zhang 50653161

Motivation of Integration

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That is what I have now. A function f(t) and quadrant on my paper. If I want to calculate the area under the function, motivation of integration appears.

We set two different number on the x-axis like l and r, making it a interval [l,r].Then we find a number in this interval,t*,and calculate the value of f(t*).

So the area approximately equals to f(t*)·(r-l)

For the deeper and more precise calculation, we divide the interval [r-l] into three equal sub-intervals like t1, t2 and t3.

Then t1=t2=t3=(r-l)/3.

And we find three number t1*, t2* and t3* which within three sub-intervals respectively, calculating their value of y-axis such as f(t1*), f(t2*) and f(t3*)

Therefore, the refined area under the function can be presented as A=f(t1*)·t1+f(t2*)·t2+f(t3*)·t3

With this trial, we can divide [l,r] into more and more sub-intervals to reach the most precise area that I want to calculate.

Of course, area only can be calculated in some functions like continuous functions, for the reason that a non-continuous function does not have closed area.

Written by Yuki Zhang(50653161) and Denise Wang(24482168)

Assignment 7

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I solved the Question 5, and I found the secret of this question.

We can know the general formula of the area is removed in each step. We treat the general formula as a series.Then the sum of the series is the area which has been removed. By the knowledge we have learned about the series. We can know the sum of this series when n approaches a large number which approaches to a constant.Then we know the Sierpinski carpet, and we can solve the question.

I think we need to transfer the questions to the knowledge we have learned.