Is area under an integral limit exact or an approximation? Suppose we need to calculate the area of the the curve $y=sin x$. Then we calculate the area  enclosed by the curve from $x=x_1$ to $x=x_2$ as $\int_{x_1}^{x_2}sin x\, dx$. Is the area calculated so exact or approximate?. In case of a linear curve such $y=ax+b$, we do get an exact value (as justified by geometry)
This confusion stems from the question whether limits are exact or not (and hence all operations related to limits).
Note: I did see other posts to see an answer, but none of them were complete or the questions were something else entirely. So this is not a duplicate.
 A: The notion of area must first be defined. Unfortunately, that is quite a tricky business. In measure theory one gives a precise notion of area for sets in $\mathbb R^2$ but it turns out that not every subset can meaningfully be assigned an area. However, it is a theorem that if a set is precisely the locus of points bounded above and below by graphs of functions $f(x)$ and $g(x)$ respectively, then the area of that set is $\int f(x)-g(x)dx$. This is then a precise justification for computing the area under the graph of a function by means of the integral. It is then a precise answer. 
Remark: You seem to be under the impression that limits are somehow imprecise or that they are approximations. This is incorrect. A limit is a number. It is not a process, nor an approximation, nor in any way imprecise. It is a very much fixed number that never ever changes. The limit $\lim _{n\to \infty }\frac{1}{n}$ is precisely $0$. It is wrong to say "it is $0$ when $n=\infty $" since $n$ is a natural number here, so $n=\infty $ is meaningless. It is wrong to say "the limit approaches $0$" or "the limit becomes $0$" or any other thing like that. The limit is simply $0$. 
A: The integral gives you 'exactly' the right area in the sense that any approximation, using rectangles or squares--which have a well defined area--will always converge on the answer you find evaluating the integral.
In other words, the way we define what the area under the curve in introductory analysis of a Riemann sum is tautologically equal to the integral.
Do you have another definition for 'the exact area' under the curve? (There's another definition called a Lebesgue integral and it turns out that for continuous functions like sin it is exactly equal to the Riemann integral.)
A: This is exactly why we have the concept of limits. 
We have a curve, $f(x) = \sin(x)$ from $0 \to \pi$ 
$\displaystyle \int_{0}^{\pi} \sin(x) \space dx = \displaystyle \lim_{||\Delta(x)|| \to 0}\sum_{i=1}^{n} f(c_i)\Delta(x)$ where $c_i$ is a respective point in the respective partition $i$.
It is saying, as the width of the Largest partition approaches $0$, what will the area be? Thinking about this, if the width of the largest partition $=0$ then that is the same as $n \to \infty$, now, if the number of rectangles goes to infinity, then the rectangles have width $dx$ making the rectangle infinitely small in width. This means the whole region is covered perfectly.
Again this is a limit, this cannot physically happen. Which is why we put it as a limit, and don't just say as $n \to \infty$, it is the limit. What would happen if $n \to \infty$
Does this help?
