That is okay, I hope you are feeling better. If not then get some rest and we can continue later or tomorrow.
Your answer is correct! We use this formula to get more.
n is the number of intervals or something like that I suppose?
How do you get this formula?
Yes, n is the number of intervals. The formula is derived in a lot of books. I am not that big on memorizing proofs or derivations so I do not recall it offhand. What is important for numerical work is the error estimate and the fact that it works!
This is known as the Trapezium Rule. Proving it can be done as follows.
It can be shown that
where U(f,P) and L(f,P) are the lower Darboux sums of f with respect to the partition P, defined by
and M_i and m_i are given by their usual definitions
Then, it is easy to see that.
A similar technique is used to show that a monotonic function on [a,b] is Riemann integrable on [a,b].
I tried that problem with larger and larger limits until it seemed that I've got the first two digits steady
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
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You used the trapezoid rule? What cutoff did you use? Did you bound the tail?
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.