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.
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 ----------
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.
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.