Calculus - Lesson 14 | Integral of a Function | Don't Memorise

[@InfinityLearn_NEET]

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[@sahilworld_explorer]

woooooo

[@egregiouslytalented2381]

u must have 10 million subscribers based on ur concept teaching .i am a 13 yr old learning calculus u guys r excellent

[@exoticcoder5365]

I wish I could have this kind of access to knowledge at your age

[@crisscringle]

Explanation for bonus question at the end :) I did the upper sum rectangles for the question, so I took the maximum x coordinate to find y f(x) = x^2, so.... 1st rectangle: Max X coord = dx y = f(dx) = (dx)^2 2nd rectangle: Max X coord = 2dx y = f(2dx) = (2dx)^2 3rd rectangle: Max X coord = 3dx y = f(3dx) = (3dx)^2 Nth rectangle: Max X coord = ndx y = f(ndx) = (ndx)^2 As you know, to get total area, you get the area of the rectangles and then add them up. So height * width; Width = dx, and height is the y coordinate of each rectangle... 1st rectangle = (dx)^2 * dx 2nd rectangle = (2dx)^2 * dx Nth rectangle = (ndx)^2 * dx (dx)^2*dx + (2dx)^2*dx + ...(ndx)^2*dx Then expand the squared terms... (I expand (dx)^2 = d^2 x^2) 1 d^2 x^2 (dx) + 4 d^2 x^2 (dx) + ... n^2 d^2 x^2 (dx) So we get a common factor of square series, which we can take out... [1 + 4 + 9 + ... n^2] (d^2 x^2) (dx) We can also simplify d^2 x^2 back to (dx)^2, so we can multiply it by dx... [1 + 4 + 9 + ... n^2] (dx)^3 We can replace the square numbers with the formula for the series of n square numbers... [(n(n + 1)(2n + 1)) / 6] (dx)^3 The width of each rectangle (dx) is the range between the limits (0 to 4) divided by the number of rectangles (n), so dx = 4/n. Sub that in... [(n(n + 1)(2n + 1)) / 6] (4/n)^3 [(n(n + 1)(2n + 1)) / 6] (64/n^3) (64n(2n^2 + 3n + 1)) / 6n^3 (32(2n^2 + 3n + 1)) / 3n^2 (64n^2 + 96n + 32) / 3n^2 Which we can separate to be... 64n^2/3n^2 + 96n/3n^2 + 32/3n^2 64/3 + 32/n + 32/3n^2 As n tends to infinity, the two last terms approach 0: 32/n and 32/3n^2 Therefore, answer is 64/3

[@nabanitachowdhury3854]

Your answer is correct but you did a mistake in writing. It would have been (64n2+96n+32)/3n2 and not (64n2+96n+1)/3n2

[@crisscringle]

Nabanita Chowdhury thanks, have corrected it :)

[@crisscringle]

OBIIOS and removed the exclamation mark 😂

[@crisscringle]

OverComplicated dude In this case, the calculations for the upper sum and the lower sum are the same. In the previous example in the video (0:20), we needed to integrate between 2 and 6. Because we were starting from 2, there was a difference between the upper sum and the lower sum. For Lower Sum, x coordinates were: 2, 2 + dx, 2 + 2dx... 2 + ndx For Upper Sum, x coordinates were: 2 + dx, 2 + 2dx... 2 + ndx The starting point of the integral determines whether the upper and lower sums will be the same. In our case of the graph of x^2, the x coordinates would be: X for Lower Sum: 0, 0 + dx, 0 + 2dx... 0 + ndx X for Upper Sum: 0 + dx, 0 + 2dx... 0 + ndx Because our integration is starting from 0, there is no actual difference once you remove the unnecessary zeroes. Both of them will end up like: dx, 2dx, ... ndx

[@hemsingh6785]

This how we should be taught early.......great work thanks teacher 😍

[@skilz8098]

The integral is 21.333... or 21 + (1/3) units cubed. Since the function is f(x) = x^2 and its indefinite integral is (1/3)x^3 + C we now use this to find the definite integral between [a,b]. W can do so by applying the upper and lower bounds of the limits of integration and then take their differences. The lower bound will evaluate to 0. (1/3)*0^3 = 0. Since we know that for any x - 0 = x; we only need to consider and evaluate its upper bound. Therefore: F(x) @ b = (1/3)*(4)^3 = (1/3)*64 = 64/3 = 21.333... or 21 + (1/3) units cubed since F(x) = (1/3)x^3 + C. What happened to the Constant of Integration? When we apply F(b) - F(a), we will have the difference between C2 and C1. Since C1 = C2 the difference will result in 0. Thus, we can easily omit it leaving us with a definite volume since we integrated an area function. f(x) = x^2 is quadratic which is an area function and f(x) = x^3 is cubic function which is a volume function. A polynomial is defined by its highest ordered exponent which is understood both Algebraically and Geometrically. Therefore, for the answer to be correct, we must apply units^3 to the finale definite value otherwise the units will be wrong giving an incorrect answer. If you have two quantities such as 10 and 30 and these values aren't scalar and we try to perform some arithmetic... we have to know their dimensionality in order to apply any operation or transformation to them. We have to convert them into the same units in order to do the appropriate arithmetic. For example, let's say we have an area A = 10 m^2 and we have a volume V = 30 mm^3. How would you do the following: A+B, A*B, A-B, A/B, etc... if we didn't specify the units. So just saying 64/3 is the integration is wrong. We must specify 64/3 units^3 for it to be correct! The units are they inches, centimeters, miles, kilometers, etc...? This is just as important as the actual value!

[@padmajamohapatra7179]

Salute you to write biggest comment of the year

[@skilz8098]

@@padmajamohapatra7179It was barely a paragraph.

[@bruhifysbackup]

In graphs, we don't write out the measurement nor the dimension were dealing with. The are under the curve doesn't have to be ³.

[@amogelangsekhu3906]

They really meant it, when they said Don't memorize. this was so helpful thank you 🔥❤️👌

[@neerxj]

thanks for removing my fear of calculus

[@deepudasari1117]

episode 1: This is so easy Episode 2: what the heck is a function? episode 3:this is getting a little confusing. episode 4: no way I can understand. this is stupid.

[@dioptre]

what grade are you in?

[@Vibranium375]

Well to me all 17 are very easy lol

[@rajatpathak6524]

Awesome work......I wish we had this kind of explanation ....when we were at school....❤️❤️❤️

[@InfinityLearn_NEET]

Thank you so much Rajat for your appreciation. We are glad that you understood the concept. For more videos, please visit our website - dontmemorise.com/

[@samehmekawy1049]

It will be 64/3 Thank alot for your videos

[@anoopaji1469]

Very useful. Thanks for the precious effort.

[@InfinityLearn_NEET]

You're most welcome, Anoop! Thanks for your support and Keep watching!! 🙂🙂

[@صفايسرى-ع6ح]

بسم الله الرحمان الرحيم ولله ملك السماوات والأرض يغفر لمن يشاء ويعذب من يشاء وكان الله غفورا رحيما

[@akavyascv0415]

You save my day mam tq so much ...

[@krishchauhan9209]

Mam please give us the solution of the problem given by you in the end of the video. Please give it in a long form just like you derived 16-8/n. Otherwise the video was 👍

[@sachinkataria8867]

Thank you so much. All videos are awesome

[@saipatil8751]

Your videos are so helpful. Keep teaching.

[@InfinityLearn_NEET]

Thank you so much Sai for your appreciation. We are really happy to hear that it was Helpful to you. We are glad that you understood the concept. You motivate us to do better. Keep watching our videos. 😊😊

[@quickSilverXMen]

This is call complete concept 🌞🌞

[@marrapujayadeep9385]

+64/3 Good teaching mam

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[@ninoreyrequirme9996]

The integral of X² is X³/3 so.. 4³/3= 64/3 and 0³/3=0 64/3+0= The area is = 64/3

[@herilmistry1263]

Thank you so much ...

[@skilz8098]

The Integral of f(x) = x^2 from [0,4] IS NOT 64/3. The Integral of f(x) = x^2 from [0,4] IS 64/3 units^3. You must specify the units! And it's not Area, In the example from the Video, it is Volume! What if we were to do some arithmetic of two integrals: F(x) from [2,5] + G(x) from [3,9]... where f(x) = 2x^2 and g(x) = 4x^3 - 2x^2 + 9? F(x) = (2/3)x^3 from [2,5] = (2/3)(5)^3 - (2/3)(2)^3 = (2/3)(125) - (2/3)(8) = (250/3) - (16/3) = (234/3) = 78 units^3 G(x) = (4/4)x^4 - (2/3)x^3 + 9x from [3,9] = { 9^4 - (2/3)9^3 + 9(9)} - { 3^4 - (2/3)3^3 + 9(3)} = (6561 - (1458/3) + 81) - (81 - (54/3) + 27) = (6561 - 486 + 81) - (81 - 18 + 27) = (6156) - (90) = 6066 units^4. So if we didn't specify the units^2, units^3, ... units^n how would we be able to determine to perform F(x) + G(x) without knowing the units? If the unit type is unknown we can generically just say units... But what if F(x) is in kilometers and G(x) is in millimeters? Now we have a problem if we don't specify the units! By specifying the units we are then able to convert from one unit to the other... Then by doing so would we then be able to perform F(x) (O) G(x) where (O) can be any operator.

[@blmandar]

I hope I've understood something.......Well, I have............Haven't I ? Great, I'm confused again.

[@carguy7480]

Same here

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[@amitpandey3250]

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[@anastasiaanautodidact9856]

Kind of same here... No examples... Just abstract content..

[@sanju6337]

Please do vedios on dynamics

[@md.zahidulislam3548]

thank you

[@Ramzaoui]

thanks from morocco

[@dhanrajambhore3441]

Area of integral : 64/3 =21.3

[@Kivhknn]

which software r u using????plxxx

[@golumandal4836]

Magnificent...

[@madurir3744]

Can u please tell me the derivations of the calculus in easy way it was little confusing for me

[@airamunazza7884]

Mam we shall divide this function in to many parts horizontally

[@dspopnrendi1725]

7:30 mam according to my calculation the answer is (n-2)16 then what after this?

[@karthickthangavel2259]

What is difference between sum of area and sum of limits

[@peterford3478]

Genius.

[@puskarbasak3712]

The integral function is approximately equal to 21.33 unit

[@aderonkeadeniyi2428]

I LIKE IT

[@aderonkeadeniyi2428]

0:02 0:03 0:03 0:03

[@sukhmeetsinghsodhi3261]

answer of the question asked is 64/3

[@padmajamohapatra7179]

64/3 is the answer to the integral

[@ninanakagawa8943]

where does the ( 16 + 8/n) come from

[@AmanSharma-we9nk]

Can anyone pls share the solution of the problem given in last with the use of lower or upper sum

[@crisscringle]

Aman Sharma In this case, lower and upper sum are actually the same f(x) = x^2, so.... 1st rectangle: Max X coord = dx y = f(dx) = (dx)^2 2nd rectangle: Max X coord = 2dx y = f(2dx) = (2dx)^2 3rd rectangle: Max X coord = 3dx y = f(3dx) = (3dx)^2 Nth rectangle: Max X coord = ndx y = f(ndx) = (ndx)^2 As you know, to get total area, you get the area of the rectangles and then add them up. So height * width; Width = dx, and height is the y coordinate of each rectangle... 1st rectangle = (dx)^2 * dx 2nd rectangle = (2dx)^2 * dx Nth rectangle = (ndx)^2 * dx (dx)^2*dx + (2dx)^2*dx + ...(ndx)^2*dx Then expand the squared terms... (I expand (dx)^2 = d^2 x^2) 1 d^2 x^2 (dx) + 4 d^2 x^2 (dx) + ... n^2 d^2 x^2 (dx) So we get a common factor of square series, which we can take out... [1 + 4 + 9 + ... n^2] (d^2 x^2) (dx) We can also simplify d^2 x^2 back to (dx)^2, so we can multiply it by dx... [1 + 4 + 9 + ... n^2] (dx)^3 We can replace the square numbers with the formula for the series of n square numbers... [(n(n + 1)(2n + 1)) / 6] (dx)^3 The width of each rectangle (dx) is the range between the limits (0 to 4) divided by the number of rectangles (n), so dx = 4/n. Sub that in... [(n(n + 1)(2n + 1)) / 6] (4/n)^3 [(n(n + 1)(2n + 1)) / 6] (64/n^3) (64n(2n^2 + 3n + 1)) / 6n^3 (32(2n^2 + 3n + 1)) / 3n^2 (64n^2 + 96n + 32) / 3n^2 Which we can separate to be... 64n^2/3n^2 + 96n/3n^2 + 32/3n^2 64/3 + 32/n + 32/3n^2 As n tends to infinity, the two last terms approach 0: 32/n and 32/3n^2 Therefore, answer is 64/3

[@kavyasangeet]

@@crisscringle you are awesome 😮

[@gauravm3333]

@@crisscringle good work man by the way thanks for explanation for this man and us

[@crisscringle]

@@gauravm3333 Glad the explanation was helpful! :)

[@shivangishivangi4267]

​@@crisscringle Thanks for the explanation,😭🙏

[@pwsk]

Wow!

[@shitalmandalia9849]

I am so curious about maths, science and physics and I want to become a aerospace engineer I am in 8th grade, I am trying my hardest to learn calculus

[@stuartsandskill270]

hey budy ,just to clesrify i am saying that aerospace engineer who is involved in the making of flight ,rocket etc.if you are really interested in physics and science,the pursue physics or maths as a core field

[@anilsharma-ev2my]

Integration continues on and on up to infinity What will we get ?

[@codecatt]

Hello teacher will you kindly tell me what video studio you're using??

[@StudentWalaGamer]

First👍👍

[@shahnirmalmtb]

64/3

[@pranavtube3302]

Solution to the problem. 21.33333....

[@blackcat5771]

1/3(x^3)

[@ryanme5645]

=3

[@adityayadav3388]

8

[@mdmia1526]

i actually can solve it

[@sukuruman9412]

Maassss

[@chitraa5715]

Attendance of students who regret for taking science in 11th 😖 👇

[@hemsingh6785]

64/3

[@wesleychan3819]

64/3