# What is the value of the integral $\int_0^\infty 0dx$?

I know that this question might be very simple, but I want to know: what is the value of following integral?

$$\int_0^\infty0dx$$

The trick with such things is always: look at the definition, and just apply it carefully.

• By definition, $$\int_0^\infty f(x)dx$$ is just $$\lim_{r\rightarrow\infty} \int_0^r f(x)dx$$, so we just need to calculate $$\int_0^r 0dx$$ for an arbitrary real $$r$$.

• But $$\int_0^r 0dx$$ is always $$0$$ (you should already be comfortable with this).

• So $$\int_0^\infty f(x)dx=\lim_{r\rightarrow\infty} \int_0^r f(x)dx=\lim_{r\rightarrow\infty}0=0$$.

The reason $$\int_0^\infty 0dx$$ may seem mysterious at first is that it feels like the old "paradox": "What happens when you add infinitely many infinitely small quantities together?" But this "paradox" is also present in integration all the time: $$\int_a^bf(x)dx$$ is intuitively "the sum of the areas of infinitely many infinitely thin rectangles." So this is a problem that we've already resolved, and the fact that $$\int_0^\infty0dx=0$$ should be no more mysterious than the general behavior of integration (in fact, it should be less mysterious).

Hint: $$\forall t>0: \int_0^t 0 \mathrm{d}x=0$$

A more intuitive angle: recall that

$$\int_a^b f(x)dx$$

can be analogized as the area between $$f$$ and the $$x$$ axis on the interval $$[a,b]$$. Take $$a=0, b \to \infty,$$ and $$f(x)=0$$. Then we have

$$\int_0^\infty 0 dx$$

or, essentially, the area between the line $$y=0$$ (the $$x$$-axis) and the $$x$$-axis, for $$x\geq0$$. In that light, it should be immediately clear there is $$0$$ area, and thus the integral is $$0$$.

By one definition, the integral is the infinite limit of the sum of the areas of rectangles whose base is $$\Delta x$$ and whose height is $$f(x_i)$$. In this case $$f(x_i)=0$$. So $$f(x_i)\Delta x=0$$. Summing these, you get, of course $$0$$ and the limit is $$0$$.