Is It Indeterminate Or Zero?

Calculus Level 1

$\large \displaystyle \int_{-\infty}^{\infty} 0 \, dx =\, ?$

0 Indeterminate

3 solutions

Hobart Pao
Nov 12, 2016

Because we have improper integral, I will rewrite as $\displaystyle \lim_{a \to -\infty, b \to \infty} \int_{a}^b 0 \, dx$ . By integrating and applying the FTC, we get

$\displaystyle \lim_{a\to -\infty, b \to \infty}\left. 0 \right|_{a}^{b} = \lim_{a\to -\infty, b \to \infty} (0-0)= \boxed{0} .$

You're right to say that the concern with this is dealing with the limits at infinity. The accurate way to deal with this, is to take $\lim_{ a \rightarrow - \infty, b \rightarrow \infty} \int_a ^ b f \, dx .$

There are some cases (like with $\int x$ ), where the $\lim_{ k \rightarrow \infty } \int _{-k}^k$ will yield a result, but the stronger definition given above will not.

Calvin Lin Staff - 4 years, 7 months ago

I have modified my solution. Look good now?

Hobart Pao - 4 years, 7 months ago

Yup Thanks!

Calvin Lin Staff - 4 years, 7 months ago

When you integrate, wouldn't it be a constant instead of 0?

Meredith Caveney - 4 years, 6 months ago

When you do the FTC, you'll notice that the constants cancel out.

Hobart Pao - 4 years, 6 months ago

梦叶
Nov 24, 2016

We are integrating nothing. There are nothing under area; thus, the integral is definitely 0.

Viki Zeta
Nov 12, 2016

$\int_{-\infty}^{\infty} 0 dx = 0 \int_{-\infty}^{\infty} 1 dx = 0 (x |^{\infty}_{-\infty}) = 0$

The potential problem I see with this is, when you take the limit as x goes to infinity, you're getting an indeterminate form when you do zero times (infinity plus infinity). Especially, if you do the order of operations, what goes in the parenthesis is done first.

Hobart Pao - 4 years, 7 months ago

Agreed.

To spell it out further, since $[ x]_{- \infty } ^ \infty = \infty$ , we now have the indeterminate form $0 \times \infty$ .

Calvin Lin Staff - 4 years, 7 months ago

Taking elementary math rules, you get '0'. Here infinity is just a large number not $\dfrac{1}{0}$

Viki Zeta - 4 years, 7 months ago

@Viki Zeta – That's the point. You have to spell out exactly what it means, as in Hobart's solution. In it's current version using $\infty$ , you run into an indeterminate form.

Calvin Lin Staff - 4 years, 6 months ago

As we know, integration of constant is variable and integration of 0 is constant. So if we Integrate 0 (don't think about limits) with respect to any variable we get a constant value right. How can u say it is zero. Pls clarify me.

Vamseekrishna Pvs - 4 years, 6 months ago

Note that we have a definite integral with given limits, and not an indefinite integral (which involves a +C).

Calvin Lin Staff - 4 years, 6 months ago

Is It Indeterminate Or Zero?

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