Guys help with integration pls😔

[unborn kelp]

So I’m self studying A Level math ahead of my class and I’m currently covering integration but I don’t understand anything T-T could someone pls explain or link a video? Tysm<3

[weary rampart]

Hi! So, integration is basically the opposite of differentiation. If differentiation is:
“Given a curve, tell me its gradient/slope at any point,” then integration is:
“Given the gradient function, tell me what the original curve was.”

And integration helps us find:

1. Mainly areas under curves

2. Accumulated quantities (like distance from velocity)

For Alevels, there are two types of integrations.

The first is Indefinite integration, which is the process of finding the original function f(x) when you're only given its derivative f'(x)

It’s called “indefinite” because you’re not given any limits (like from
a to b). Instead, you get a general formula that includes a constant +c

why +c is because when we reverse the differentiate process (integrate), we don’t know what constant might have been there, so we just write +C.

The second is definite integration, which is basically finding the area under the curve between two x-values

normally, it looks like this, and means 'find the area between the curve f(x), the x-axis, and the vertical lines x=a and x=b' .

Still, we use the same rule as indefinite integrals here, but when calculating the final result we don't need to plus c. You can have a go by yourself, and will find that c cancels out

so, step1 is to integrate the function, step two plug in b and a separately and calculate the difference, ie f(b) - f(a)

A simple example would be

i.e, We want to find the area under the curve y=2x, from x=1 to x=3

[weary rampart]

we integrate using this formula, and we get x^2

and then we plug in 3 and 1 separately, calculating 3^2 - 1^2 = 8

and this is the final answer :)

These are basically the fundamentals, lemme know if you have any further questions :)

[unborn kelp]

Thank you so much Ari!!

So if I understand correctly, integration can be used to calculate probability? Or am I wrong

[weary rampart]

mm yess you are correct!

we need to enter a PDF here which is probability density function

it's a curve, and the actual probability is the area under the curve

Because integration literally means “Add up all the tiny areas under a curve from point A to point B”

So if you want to know, let's say 'what’s the probability that a person is between 160cm and 170cm tall?'

you write sth like this

f(x) is the pdf

The integral gives the area under the curve

And that area = probability

[unborn kelp]

Thanks a lottt 🤍

[weary rampart]

you are welcome!! lemme know at any time if you need help with anything else🥺💗

[silk niche]

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[steep shard]

Just a question,
let o = σ and u = μ
If the empirical rule is difficult to apply in some scenario… to find the probability of something, would you integrate this function between the designated bounds? Does this integral have an “easy” elementary solution? If you answered this already my apologies

[weary rampart]

Hii, thank you for your question! I took a little time to think about it, and my idea is that youd still compute it in the same way, tho there is no way to integrate this function exactly using basic calculus. The antiderivative doesn't exist in terms of elementary functions, and I think we need other tools like erf or z tables, and this type of functions are most likely calculated by calculators :)

I haven't dived really deep into this field, so pls feel free to share if you find out any other solutions to the problem!!

[coarse cargo]

How does gradient relate to area tho?

[steep shard]

Differentiation finds out the instantaneous rate of change anywhere along a function

Integration reverses this process by finding the total accumulation along this function. In other words, it finds the net area. If you want a better, more in-depth explanation, try this video https://youtu.be/FnJqaIESC2s?si=Zvx8evie_eNnu3O3

@coarse cargo

[coarse cargo]

thanks ill check it out

[wraith basin]

This one's a real great video and series ngl

[raven urchin]

i found out this website recently, hope that can help
(https://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx)