MHB Understanding the Meaning of Integration in Math

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Integration in mathematics refers to the process of finding the integral of a function, which represents the accumulation of quantities and the area under a curve. It is a fundamental concept in calculus, often requiring extensive study to fully grasp its applications and implications. The integral can be classified into definite and indefinite integrals, each serving different purposes in mathematical analysis. Understanding integration involves not only the computation of integrals but also the interpretation of their results in various contexts. Mastery of this topic typically spans multiple semesters in college-level courses.
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what is The meaning of integration?
 
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You understand, I hope, that it can take whole one and two semester college courses to answer that question?
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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