hey guys,
I was reading a calculus text book and I'm on the section of Riemann's integral, now the example my text uses is, to find the area under the curve f(x)=x²+1, now the book explains the solution like this
![latex2png.2.png latex2png.2.png](https://www.electro-tech-online.com/data/attachments/37/37652-ccb50094f34fb31c352f52087d414f57.jpg)
It's later simplified like this:
![latex2png.3.png latex2png.3.png](https://www.electro-tech-online.com/data/attachments/37/37647-1aece678f4c1439b9d287006f0cbd563.jpg)
![latex2png.4.png latex2png.4.png](https://www.electro-tech-online.com/data/attachments/37/37648-f7f085b9e7bfc4937f6476250210586f.jpg)
NOTE: The curve occurs between x = 0 and x = 3
Now the part that I don't understand is the part where the book tells you that the sum of the first n squares is equal to:
![latex2png.5.png latex2png.5.png](https://www.electro-tech-online.com/data/attachments/37/37649-ba9ddd960afd10926862d7f713533366.jpg)
Is this a general formula that the author knows from previous experience or did he get this by simplifying any of the latter formulas? The reason I want to know this is because the formula is then written like this:
![latex2png.7.png latex2png.7.png](https://www.electro-tech-online.com/data/attachments/37/37651-965bc12e6c3f3188b0bbb6f499b0bd7a.jpg)
is replaced with n since if you add 1 n times you get n
So if it the author didn't get this from simplifying any of the latter formulas, what would have happened if we were looking at a different function (eg. f(x)=x³-2x)
I was reading a calculus text book and I'm on the section of Riemann's integral, now the example my text uses is, to find the area under the curve f(x)=x²+1, now the book explains the solution like this
![latex2png.2.png latex2png.2.png](https://www.electro-tech-online.com/data/attachments/37/37652-ccb50094f34fb31c352f52087d414f57.jpg)
It's later simplified like this:
![latex2png.3.png latex2png.3.png](https://www.electro-tech-online.com/data/attachments/37/37647-1aece678f4c1439b9d287006f0cbd563.jpg)
![latex2png.4.png latex2png.4.png](https://www.electro-tech-online.com/data/attachments/37/37648-f7f085b9e7bfc4937f6476250210586f.jpg)
NOTE: The curve occurs between x = 0 and x = 3
Now the part that I don't understand is the part where the book tells you that the sum of the first n squares is equal to:
![latex2png.5.png latex2png.5.png](https://www.electro-tech-online.com/data/attachments/37/37649-ba9ddd960afd10926862d7f713533366.jpg)
Is this a general formula that the author knows from previous experience or did he get this by simplifying any of the latter formulas? The reason I want to know this is because the formula is then written like this:
![latex2png.7.png latex2png.7.png](https://www.electro-tech-online.com/data/attachments/37/37651-965bc12e6c3f3188b0bbb6f499b0bd7a.jpg)
![latex2png.6.png latex2png.6.png](https://www.electro-tech-online.com/data/attachments/37/37650-6bea6d14860d5541abe7c9534595935a.jpg)
So if it the author didn't get this from simplifying any of the latter formulas, what would have happened if we were looking at a different function (eg. f(x)=x³-2x)