Step by step
The question
Question: find the integral from 1 to infinity of 1/x².
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Why it is improper
Draw the curve.
The area under it goes on to the right for ever: the top limit is infinity.
Infinity is not a number we can put in.
So we stop at a letter, t, and then let t grow without end.
2 Use a limit
The integral is the limit of the integral from 1 to t, as t goes to infinity.
3 Integrate
Write it as x to the minus 2.
The power rule: x to the minus 2 becomes x to the −1, divided by −1.
4 Let t go
Put in t, then take away the same with 1.
As t grows, minus 1/t goes to 0: a number over something huge is tiny.
The t part goes to 0.
5 Decide
The limit is a number: the integral converges to 1.
An area that goes on for ever, but adds up to 1!
Careful!
Never put infinity in like a number.
We let t grow and watch where the answer goes.
6 Check
Let's check with a calculator: the integral up to t = 1000 is about 0.999.
Up to a million, about 1.
Closer and closer to 1.
Right!
7 Pick the choice
Our answer is 1.
That is choice B.
8 Why not the others
A is infinity.
Wrong: 1/x² shrinks fast enough: the area is finite.
C is 0.
Wrong: the top end gives 0, the bottom end gives 1.
D is −1.
Wrong: the sign: −1/x from 1 to ∞ is 0 − (−1).
Let's look back.
First: why it is improper.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Then: check.
Then: pick the choice.
Last: why not the others.