Why is the integral defined as the limit of the sum $\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x$?
116622
What's the "limit" in the definition of Riemann integrals?
60792
Limit of divided integrals
13287
Let $\int_{- \infty}^{\infty} f(x) dx =1$. Then show that $ \int_{- \infty}^{\infty} \frac{1}{1+ f(x)} dx = \infty.$
104394
What does $\int_{a}^{b} f(x)$ mean?
111279
The value of $\lim_{n \to \infty}\int_{0}^{1}nx^ne^{x^2}dx$
54304
How to make sense out of the $\epsilon-\delta$ definition of a limit?
110208
limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$
134787
How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$
5402
Prove or the disprove the existence of a limit of integrals
107351
Evaluate the improper integral $\int_{0}^{\infty}{f(x)-f(2x)\over x}dx$, where $\lim_{x \to \infty} f(x) = L$
88226
Limit of a stochastic integral
105860
Evaluate the limit of $(n+1)\int_0^1x^n\ln(1+x)\,dx$ when $n\to\infty$
118056
Limit of a Wiener integral
74454
Sophomore's dream: $\int_0^{1} x^{-x} \; dx = \sum_{n=1}^\infty n^{-n}$
149068
Prove: Assume $\lim_{n\to\infty}(a_n)=a, \lim_{n\to\infty}(b_n)=b$ then $\lim_{n\to\infty}(a_n+b_n)=a+b$
32480
Limit of integration can't be the same as variable of integration?
55694
Why is the upper Riemann integral the infimum of all upper sums?