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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,

etc.

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

QUESTION: Hi Scott,

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.

regards.

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 ----------

QUESTION: Hi,

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.

Calculus

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