TITLE: Defining the area under an oscillating function
QUESTION [6 upvotes]: I was curious about taking a definite integral of an oscillating function. 
For example,
$$\lim_{a\to 0} \int_a^1 \sin \frac1x\,dx$$
I know that there is some area under the function, but since it oscillates infinitely is it possible to define it? Do you use the limit superior and the limit inferior? I know it is probably possible (somehow) to take the antiderivative, FTC, etc. but I am wondering what this really intuitively means, given that this function oscillates (so we can't really "see" the area under the function).

REPLY [2 votes]: After robjohn's answer, using special functions $$I=\int \sin\left(\frac 1x\right)~dx=x \sin \left(\frac{1}{x}\right)-\text{Ci}\left(\frac{1}{x}\right)$$ where appears the cosine integral. So,$$J=\int_a^1 \sin\left(\frac 1x\right)~dx=\text{Ci}\left(\frac{1}{a}\right)-a \sin \left(\frac{1}{a}\right)-\text{Ci}(1)+\sin
   (1)$$ and $$\lim_{a\to0^+}\int_a^1\sin\left(\frac1x\right)\,\mathrm{d}x
=\sin (1)-\text{Ci}(1)\approx 0.5040670619$$