WEBVTT

1
00:00:00.000 --> 00:00:03.715
Here is the problem: e to the power negative 2x.

2
00:00:04.165 --> 00:00:09.320
Evaluate the improper integral of e to the power negative 2 x) from 0 to inf.

3
00:00:10.120 --> 00:00:19.803
This is an improper integral. It is improper because a limit of integration is infinite. It is defined as a limit, and it converges only if every limit is finite.

4
00:00:20.253 --> 00:00:28.334
Exponential rule: the integral of eᵘ du equals eᵘ (divide by the inner slope).

5
00:00:28.783 --> 00:00:29.828
Antiderivative.

6
00:00:30.278 --> 00:00:32.646
Evaluate the piece from 0 to inf.

7
00:00:33.446 --> 00:00:38.090
Add the pieces. Every limit is finite, so the integral converges.

8
00:00:38.890 --> 00:00:43.546
So the answer is Converges: 1 over 2 approximately 0.5.

9
00:00:43.996 --> 00:00:54.747
An improper integral replaces the problem point with a variable, integrates normally, and then takes a limit. A finite limit means the infinite region still has finite area.

10
00:00:55.547 --> 00:00:59.344
Antiderivative verified by differentiation, which checks out.

11
00:00:59.794 --> 00:01:05.576
Each limit of the antiderivative was computed and checked numerically by the limit engine, which checks out.

12
00:01:06.026 --> 00:01:16.220
Truncated numerical estimate (stopping 10 to the power minus 6 from asymptotes and at plus or minus 1000 for infinite limits): 0.5.