Question

True or False:\[\Large\int f(x)dx = \lim_{\Delta x \rightarrow 0}\Delta x\sum_{n=0}^{x}f\left(\dfrac{n}{\Delta x}\right)\]

If false, what's the correct one?

Answer 1

That's false.

Answer 2

maybe\[\Large\int_{0}^{x} f(x)dx = \lim_{\Delta x \rightarrow 0}\Delta x\sum_{n=0}^{x}f\left(n{\Delta x}\right)\]

Answer 3

still false

Answer 4

The limit of a sum definition of an indefinite integral is:\[\bf \int\limits_{a}^{b}f(x) \ dx=\lim_{n \rightarrow \infty }\sum_{i=0}^{n}f(x_i) \Delta x\]

Answer 5

sorry i meant to say definite integral not indefinite.

Answer 6

Basically what it means is that if the function f(x), has its interval (a,b) divided into n subintervals each of length delta x, and i'th value is picked with in each subinterval, then as the number of those subintervals approaches infinity, delta x approaches 0, and the approximation of the area under the curve gets more and more and more accurate to the point where its exactly the area under the curve.

Answer 7

I didn't know the indefinite integral had a definition in terms of summations :3

Answer 8

false