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Calculus/Regaridng Integral


QUESTION: Hi Scotto,

Let me put my understanding of Integral first. I thought Integral (I could not find the symbol in my computer) of dx means addition of minute dx to form x. I would like to know how could I understand Integral of x/2 dx. I know that the answer is x^3/3. But I wanted to know how addition of minute x/2 led to x^3/3.

Please explain.

ANSWER: It can't.  If it appears, there is a misprint in the book.
In the following answers, I assume that we have the integral, the function given,
and a dx at the end.  Also, we need a + C with no limits.

If the x/2 was really x^2, it was done correctly.

If the problem is to integrate x/2, the answer is x^2/4 + C.

If the answer is x^3/3 + C, the function integrated was x^2.

The answer to any integral of powers of x always raises the power of x by only 1 and then divides by x.  For example,
the integral of x is x^2/2 + C,
the integral of x^2 is x^3/3 + C,
the integral of x^3 is x^4/4 + C,

---------- FOLLOW-UP ----------


Thanks for you reply. Yes it is a mistake in my part. ok.. integral of x is x^2/2 + C. I want to understand the meaning of this. If I add minute xdx, how did it led to x^2/2. For example lets say x=2, then I would think integral of xdx as addition of 2dx,2dx,...and so on.. but the answer could not be x^2/2. Please clarify how addition of minute 2dx led to x^2/2.


ANSWER: Sorry about the time, but I couldn't get on the PC lately.

That just the way calculus goes.  It always adds one to x when integrating.
To find the are of a rectangle, the height is multiplied by the width, which is x.
To find the area beneath a line, the height is multiplied by infinitely small values of length.
That is where the dx and integration comes from.

Let's look at the integral of x^2 from 2 to 5.
If we divide the area into n rectangles, each of them would have a width of 3/n.

The left hand side of each of these rectangles would be sum(k=1 to n)[(3/n)[(k-1)/n)]^2].
The right hand side of each of these rectangles would be sum(k=1 to n)[(3/n)((k/n)^2).

This gives the following:

x^2, from 2 to 5          

One rectangle   width   x   y   left   right
     2   4   12      12
  3   5   25      75   75

Two rectangles   width   x   y   area
     2   4   6      24.375
  1.5   3.5   12.25   18.375   18.375   55.875
  1.5   5   25      37.5   

3 rectangles   width   x   y   area
     2   4   4      29
  1   3   9   9   9   50
  1   4   16   16   16   
  1   5   25      25   

4 rectangles   width   x   y   area      
     2   4   3      31.40625
  0.75   2.75   7.5625   5.6719   5.67188   47.15625
  0.75   3.5   12.25   9.1875   9.1875   
  0.75   4.25   18.0625   13.547   13.5469
  0.75   5   25      18.75   

---------- FOLLOW-UP ----------


Thanks for your reply. Why we always express the result of integration appended with C? What does C denote?

Since integrals are used to find the area beneath a curve, they need an upper and lower limit to do so.  If there are no limits, a +C is put in to denote that some constant value should be added.

If we were integrating f(x) = 6x^2, the answer would be F(x) = 2x^3 + C.
If it were done from 2 to 4, the answer would be F(4) - F(2) =
2(4^3) - 2(2^3) = 2(64) - 2(8) = 128  - 16 = 112.

Since there are limits, C is not put in.  If it was put in, the answer would be
128 + C - 1 - C, so the C's would cancel. So the C's cancel when limits are used,
so they aren't even included.  


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