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I took Calc I course and currently studying Calc II. I'm pretty good at integration and differentiation in terms of mathematically solving equations, but I don't truly understand the concepts.

When I apply integration rules to a solve a problem it just feels like magic to me, so I don't really get what's going on. For instance, if try to a solve an unfamiliar application problem I won't really be able to solve it. I feel like I'm just a useless calculator. At the beginning, When I started to learn about differentiation I had a pretty good understanding like finding the instantaneous rate of change, and understood the relation distance => velocity => acceleration (and why it works that way). Now I lost track.

I want to be able to use these concepts creatively in real-world applications. Can you suggest a book or a give me an advice on how to truly understand these concepts?

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  • $\begingroup$ Were you given any definition of definite integrals? ​ If yes, do you still remember it? ​ If yes to both, what is it? ​ (There are several possibilities.) ​ ​ ​ ​ $\endgroup$ – user57159 Feb 7 '17 at 10:45
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    $\begingroup$ @someone: $\int_0^{10}(5-x)\,dx=0$. What area is zero? $\endgroup$ – Martin Argerami Feb 7 '17 at 11:01
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    $\begingroup$ @MartinArgerami 5x-x^2/2 solving from 10 to 0 would be (0-0) = 0 I guess they would cancel each other over this interval. I tried to graph it there is a positive area from 0 to 5 and a negative area from 5 to 10 with the same size so the result is zero. $\endgroup$ – someone Feb 7 '17 at 11:11
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    $\begingroup$ See Fundamental theorem of calculus : from the derivative function (plotted gives the rate of change of the "magnitude") you integrate and recover the "original" function. $\endgroup$ – Mauro ALLEGRANZA Feb 7 '17 at 11:20
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    $\begingroup$ @someone: I will, later today. $\endgroup$ – Martin Argerami Feb 7 '17 at 12:19
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I really suggest PatrickJMT. He has a lot of categorized and very helpful videos and can help fill in gaps in knowledge so you have a good foundation. It certainly helped get through IB maths HL Calculus - options and (first year) University Calculus.

Applications, you can look at optimization, and a lot of physics and modeling aspects depending on your interests. Related rates may give you more insight and problems with volume, area etc.

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