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Jess
06/12/2025
Maths
Integration
145
•
6 Dec 2025
•
Understanding Integration: Calculating Areas under Curves
Jess
@jess_eqxp
1 / 3
Understanding Integration Basics
Integration is like being a mathematical detective - you're given the rate of change and need to work backwards to find the original function y. It's the complete opposite of differentiation, which makes it incredibly useful once you get the hang of it.
Here's the key rule you absolutely need to memorise: integrating x^n gives x^/. For example, if you're integrating x², you increase the power to 3 and divide by 3, giving you x³/3. Dead simple once you practise it a few times.
The tricky bit is remembering the constant of integration (c). Since differentiating any constant gives zero, when you integrate, you need to add '+c' because there could have been any constant in the original function. Don't forget this - it'll cost you marks!
Quick Check: Remember that ∫2x dx = x² + c, which reads as "the integral of 2x with respect to x"
Advanced Integration Techniques
The brilliant thing about integration is that the formula works for negative and fractional powers too - not just positive integers. The same rule applies: increase the power by 1, then divide by the new power. The only exception is when n = -1, which would give you division by zero.
Square roots and fractions become much easier when you write them as powers. For instance, √x becomes x^(1/2), and 1/x³ becomes x^(-3). Then you can use the standard formula without any stress.
Definite integrals have limits (those numbers at the top and bottom of the integral sign). You integrate normally, then substitute the upper limit minus the lower limit. This gives you an actual number rather than an expression with +c.
Pro Tip: Always rewrite roots and fractions as powers before integrating - it'll save you loads of confusion
Calculating Areas Under Curves
Definite integrals are absolutely brilliant for finding areas under curves. The process is straightforward: integrate the function, pop it in square brackets with the limits, then substitute the upper and lower values before subtracting.
Here's where it gets interesting - if your curve dips below the x-axis, the integral becomes negative. Since area can't actually be negative in real life, you'll need to make it positive. This happens because the maths treats "below the axis" as negative area.
To find where a curve crosses the x-axis, set the function equal to zero and solve. These crossing points often become your integration limits when calculating total areas. It's like finding the boundaries of the region you're measuring.
When calculating total area that includes regions both above and below the x-axis, you'll need to split your integral at the crossing points and add the absolute values together.
Remember: Negative integrals just mean the area is below the x-axis - make them positive for actual area calculations
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This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
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very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
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I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
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THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
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This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding Integration: Calculating Areas under Curves
Jess
@jess_eqxp
Integration is essentially differentiation in reverse - it's one of the most powerful tools in A-level maths. Think of it as working backwards from a gradient to find the original function, plus it helps you calculate areas under curves.
Understanding Integration Basics
Integration is like being a mathematical detective - you're given the rate of change and need to work backwards to find the original function y. It's the complete opposite of differentiation, which makes it incredibly useful once you get the hang of it.
Here's the key rule you absolutely need to memorise: integrating x^n gives x^/. For example, if you're integrating x², you increase the power to 3 and divide by 3, giving you x³/3. Dead simple once you practise it a few times.
The tricky bit is remembering the constant of integration (c). Since differentiating any constant gives zero, when you integrate, you need to add '+c' because there could have been any constant in the original function. Don't forget this - it'll cost you marks!
Quick Check: Remember that ∫2x dx = x² + c, which reads as "the integral of 2x with respect to x"
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Advanced Integration Techniques
The brilliant thing about integration is that the formula works for negative and fractional powers too - not just positive integers. The same rule applies: increase the power by 1, then divide by the new power. The only exception is when n = -1, which would give you division by zero.
Square roots and fractions become much easier when you write them as powers. For instance, √x becomes x^(1/2), and 1/x³ becomes x^(-3). Then you can use the standard formula without any stress.
Definite integrals have limits (those numbers at the top and bottom of the integral sign). You integrate normally, then substitute the upper limit minus the lower limit. This gives you an actual number rather than an expression with +c.
Pro Tip: Always rewrite roots and fractions as powers before integrating - it'll save you loads of confusion
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Calculating Areas Under Curves
Definite integrals are absolutely brilliant for finding areas under curves. The process is straightforward: integrate the function, pop it in square brackets with the limits, then substitute the upper and lower values before subtracting.
Here's where it gets interesting - if your curve dips below the x-axis, the integral becomes negative. Since area can't actually be negative in real life, you'll need to make it positive. This happens because the maths treats "below the axis" as negative area.
To find where a curve crosses the x-axis, set the function equal to zero and solve. These crossing points often become your integration limits when calculating total areas. It's like finding the boundaries of the region you're measuring.
When calculating total area that includes regions both above and below the x-axis, you'll need to split your integral at the crossing points and add the absolute values together.
Remember: Negative integrals just mean the area is below the x-axis - make them positive for actual area calculations
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
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Most popular content: Power Rule for Integration
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Quotes from every main character
English Literature10
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
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4.8/5
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user