I always had a problem with Integral Calculus in my high school and was happy that I don’t need to study integrals anymore. but your explanation was awesome. I now really know what integration means! Thanks.
Congrats on a great explanation. I think I have always been able to envision integrals properly–because I could always see when it’s to be applied–but I never figured out that it was generalized multiplication that I was seeing. Of course, I am kicking myself at how seemingly obvious the integral’s identity is. 
@Nobody: Thanks, glad you liked it! I’m the same way, I used integrals for a long time without really getting what they were about. They seemed so disconnected, or like a special operation.
Well written! Another angle on understanding integration is to consider the average value of a function. Then the area of a circle becomes the average radius times the circumference, or alternatively the average circumference (which is pi * r) times the radius.
@CJ: Great point. Another way to see the integral is multiplying the average value of a function over the interval you’re considering.
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A most enjoyable read. This is probably your best article yet!
I took Calculus BC back in high school, but I was forced through all the material at breakneck speed. I scored great on the test but was left feeling it was all a waste of time.
Now, thanks to you and other sites and books, I’m combing through the material again in an attempt to truly understand it. Incredibly- it’s actually fun! I hope to pass my enjoyment along as a tutor someday.
Thanks for the work you’ve put into these articles. Your belief seems to be that anyone can understand something if it’s only explained properly. If I’m correct in that, it’s a fantastic attitude to hold.
@Mr. ag: Thank you, glad you liked it! I’m definitely a believer that any subject can be understood if explained properly. Our ‘basic’ reading, writing, arithmetic and algebra skills were once considered extreme specialties a few hundred years ago. Our brains are capable of so much if subjects are presented well :). Appreciate the comment!
Great explanation.i’ve really enjoyed your articles.could you please explain how the formulas for integration came into existence(like integral of cosx=sinx).i was eagerly looking forward your next article in this topic.
finally after all these years i am beginning to make sense of the purpose of integration in real life. well not really. its just as confusing 
but your article is definitely helping me see calculus as something beyond 50 marks worth of questions that it was in the 12th standard.
you are gifted at explaining things. its truly a gift to be able to to break things down to simple levels and explain it to people. as a novice teacher i know how tough it is to teach!! keep at it !!!
This is actually how my math professor explained it. He gave us the theory after explaining it in this way. I also got a double dose as my physics professor explained it the same way and I took physics and calculus as co-reqs.
As a side effect, I always wondered why others said calculus was so hard. Now I guess I understand where they are coming from.
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I’m surprised you used ‘d’ for Delta instead of the standard Greek letter `delta’.
^Blame Leibniz.
@vishwanath: Thanks for the encouragement! Sometimes I can only reach insights after banging my head against the wall, I’m really happy you found it useful!
@sil-chan: That’s awesome, I wish more teachers taught it that way! Only showing the formal book definition tends to confuse students.
@William, Nobody: Yeah, I didn’t want to use the actual Greek letter – too hard to type in plain HTML :).
What the… Integration as analogous to multiplication?? I never thought of that. How’d you ever come to that “aha!” moment
@love-hate: Glad it helped! I intuitively see integration as a bunch of little additions, and multiplication is like a bunch of additions also. Also, the units end up being the same (integral x dx has units x squared, x times x has units x squared).