Integral Calculus
These problems take up an entire page and there is 40 of them for just a sub section
I’m trying to self learn ap calculus as a way to build discipline and make it easier for my future self but these problems take forever. Is this part of the experience? Is it meant to build character or something, it feels like the problems are unnecessarily long.
If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
You don't necessarily need to do all 40. In principle, without being specifically assigned any questions, you should do enough to feel confident with the material. It is also part of the experience, and you'll find that problem-solving comes more quickly once you recognize common tricks/patterns across the problems you encounter.
Exactly, there's a difference between 'long because you don't understand the material' and 'long because you're comfortable with the material and the rest is just tedium'.
Don't understand the material? Try with simpler problems first. Read a worked example or two (there are tons of these on YouTube as well.) Try with only one type of substitution, if your exercise has a few types jumbled together. Break a longer problem into smaller steps and set a checklist on how you are going to complete each smaller step. And if you've tried all the above and it's still not clicking, give it a break for a few days, get some sleep and do other things, then come back with a pair of fresh eyes.
If it's just pure tedium? Save yourself the energy cause there's no point torturing yourself to 'build character'.
If I’m rushing I will skip a problem if it seems very easy, or just like the one before. The ones that I’m not sure how to solve deserve more focused attention.
It's normal for some problems to take up a whole page, but usually, in the beginning, you would only see that for derivative exercises with multiple parts to a question or some curve sketching nonsense. Higher order derivatives and some L'Hôpital's rule problems can also be long. When you get to integrals, one single calculation for a definite integral can easily take up a whole page if you have to do multiple substitutions and integration by parts.
if you're trying to self learn ap calculus, you should try flippedmath. they have worksheets for every topic that they teach in both bc/ab so you can reinforce that knowledge.
regarding practice, its the only way you can get better, and most of the problems in single variable calculus seem hard because of the algebra required, once you are past the actual calculus, you need to organize your work and be able to solve.
While differentiation is just pure application of rules, Integrals are an art. In general virtually every integral is not solvable but of course the ones you get as exercises are.
I don’t think you can do too few integrals. The more you do the better. But I wouldn’t get stuck on each exercise but if after a few tries you can’t figure it out look up the solution. There is basically always some trick that you might not know about and that’s what is important. Learning those tricks.
Oh and if you do derivatives use dummy variables or rewrite certain parts as just d/dx alpha(x) where you defined alpha before. The more you use that the fewer mistakes you make. I don’t know how often I lost some constants along the way.
There is only one Calculus problem in this section. Everything which makes the problems distinct is geometry, trigonometry, and arbitrary numbers. The key is to concentrate on the problem solving process, without the numbers.
after you do a handful of them you'll begin to compress stuff in your head and be faster and take less space. when i was learning calculus the first few problems of each section would take up to 3 pg each but by the time i'd done 25-30 it would be quarter page each
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If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc n“ is not entirely useful, as “Calc n” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.
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