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Our teacher asked us to solve some integrals and when I asked if it we are solving for the net area or signed area, they said that they are not asking for the area.

This got me thinking. Consider the integral $$\int_{-1}^{1}\frac{1}{x^{2}}\,dx.$$

We see that $$\int\frac{1}{x^{2}}\,dx = -\frac{1}{x} + C.$$ Then, $$-\frac{1}{x}\bigg|_{-1}^{1} = -2.$$ I know that $$\lim_{x \to 0}\frac{1}{x^{2}} = +\infty.$$ This means that the integral should be divergent. Without considering areas, is this value meaningful?


Edit: The 'area under the curve' means either the net or total area.

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  • $\begingroup$ It is because questions like this deserve careful answers that there is significant development of the theory of integration, and that careful definitions are given of the limiting processes involved. See, for example en.wikipedia.org/wiki/Riemann_integral for the Riemann Integral. The general theory of integration takes into account signed areas (and volumes etc). The integral of a positive function cannot be negative (it may be undefined). Integration is not just about "area" and some definitions will allow you to integrate functions which have no intuitive notion of area. $\endgroup$ May 11 at 15:17
  • $\begingroup$ By positive function, you mean functions that are positive in the entire domain? $\endgroup$
    – soupless
    May 11 at 15:21
  • $\begingroup$ @MarkBennet "some definitions will allow you to integrate functions which have no intuitive notion of area": Interesting comment. The wikipedia article that you cited referred to alternative definitions of the integral, besides the Reimann integral. Is this what you are referring to? $\endgroup$ May 11 at 15:23
  • $\begingroup$ $\frac 1{x^2}$ is everywhere non-negative, and undefined at $x=0$. Any method which does integrate this function (with an appropriate fix at zero) will give a positive answer. $\endgroup$ May 11 at 15:27
  • $\begingroup$ @user2661923 So one can integrate the function which is equal to $1$ at rational values and $0$ at irrational ones over the interval $[0,1]$ for example. The value comes out at zero - but if you have a natural intuition for the area beneath a function which is nowhere continuous you are doing rather better than me. $\endgroup$ May 11 at 15:30
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Definite integrals (of the kind studied in calculus classes--that is, Riemann integrals) are always tied to Riemann sums, and Riemann sums can be thought of as representing (signed) areas. So while your teacher may not have been explicitly asking for an area, the answer could be interpreted as a (signed) area.

There are two mistakes in what you have written:

  1. You seem to be applying the fundamental theorem of calculus to evaluate $\int_{-1}^1 (1/x^2)\,dx$. But the fundamental theorem of calculus only applies to continuous functions, and $1/x^2$ is discontinuous at 0. So your calculation that the value of the integral is -2 is incorrect; you cannot use the fundamental theorem of calculus to evaluate this integral. (Of course, $1/x^2$ is always positive, so the fact that your answer is negative should be a tipoff that something is wrong.)
  2. You are right that $\lim_{x \to 0} 1/x^2 = \infty$. But that doesn't tell you that the integral diverges, it just tells you that it is an improper integral, which means that it should be evaluated like this: $$ \int_{-1}^1 \frac{1}{x^2}\,dx = \int_{-1}^0 \frac{1}{x^2}\,dx + \int_0^1 \frac{1}{x^2}\,dx = \lim_{a \to 0^-} \int_{-1}^a \frac{1}{x^2}\,dx + \lim_{b \to 0^+} \int_b^1 \frac{1}{x^2}\,dx. $$ Whether the integral converges or diverges is determined by whether or not the limits on the right-hand side above are defined.
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  • $\begingroup$ The second item that you mentioned really makes sense. I will try to do that soon. $\endgroup$
    – soupless
    May 11 at 15:30
  • $\begingroup$ Further to point $2$, in some cases a function that diverges at $0$ has a unique well-defined integral on $[-1,\,1]$ no matter how you define integrals, e.g. $|x|^{-1/2}$. $\endgroup$
    – J.G.
    May 11 at 15:31

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