Step by step
The question
Question: find the integral from 0 to 9 of 2 over √x.
1 Why is it improper?
Draw the curve.
At x = 0 the curve shoots up for ever: 1/0 has no value.
The area is not closed at the top.
We cannot put in 0.
So we start at a letter, t, just right of 0, and then let t shrink to 0.
2 Use a limit
The integral is the limit of the integral from t to 9, as t goes to 0 from the right.
3 Integrate
The power rule: x to the minus 1/2 becomes x to the 1/2, divided by 1/2.
4 Let t go
Put in 9, then take away the same with t.
As t shrinks to 0, 4 √t goes to 0.
The t part goes to 0.
5 Decide
The limit is a number: the integral converges to 12.
An area that goes on for ever, but adds up to 12!
Careful!
Never put 0 in like a number.
We let t shrink and watch where the answer goes.
6 Check
Let's check with a calculator: the integral from t = 0.001 is about 11.8735.
Already close to 12.
Right!
Let's look back.
First: why is it improper?.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Last: check.