Why Doesn’t The Integral Equal Zero?

An interesting paradox in mathematics and calculus

Why?

Why can’t we say the integral of f(x) = x from minus infinity to positive infinity = 0?

Yeah, why is that exactly?

Let’s look at the algebra first.

Algebraically, we see that upon integration, we arrive at infinity minus infinity.

An important rule in mathematics is that we can’t treat infinity as a number. Just as dividing by zero would result in chaos, meddling with infinity the wrong way brings us a whole range of paradoxes.

What about a graphical approach? Our intuition seems to tell us otherwise.

You might be wondering that if we think of the integral as the area under the line, it might be tempting to conclude that since the area to the right of the origin is symmetric to the area to the left of the origin, they should cancel out and the total area should be zero.

But you’d be wrong!

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Shifted Up?

Consider the line y = x + 1. One way to interpret it is to see it as the line y = x but shifted upwards by one unit. Therefore, the area should increase.

Or Shifted Left?

However, we can also see that as a shift to the left by one unit. Therefore, the area should remain the same!

So which is it?

In general, using infinity as a true value will always get us into trouble, mathematically!

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Happy reading,

Barry 🍩