Worked examples
The question
Question: find the integral from 1 to infinity of 1/x².
∫ from 1 to ∞ of 1x2 dx =
1 Why is it 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.
∞ is not a number: use t, then let t → ∞
2 Use a limit
The integral is the limit of the integral from 1 to t, as t goes to infinity.
∫ from 1 to ∞ of 1x2 dx = lim t → ∞ ∫ from 1 to t of 1x2 dx
3 Integrate
The power rule: x to the minus 2 becomes x to the −1, divided by −1.
= lim t → ∞ [ −1x ] from 1 to t
4 Let t go
Put in t, then take away the same with 1.
= lim t → ∞ (−1t − (−1))
As t grows, minus 1/t goes to 0.
−1t → 0
The t part goes to 0.
= 0 − (−1)
5 Decide
The limit is a number: the integral converges to 1.
An area that goes on for ever, but adds up to 1!
= 1
Careful!
Never put infinity in like a number.
We let t grow and watch where the answer goes.
Careful: not put in ∞: −1∞ ✗
6 Check
Let's check with a calculator: the integral up to t = 1000 is about 0.999.
Already close to 1.
Right!
t = 1000: 0.999
Let's look back.
First: why is it improper?.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Last: check.
Answer Careful: not put in ∞: −1∞ ✗
▶ Watch it solved
The question
Question: find the integral from 1 to infinity of 1 over √x.
∫ from 1 to ∞ of 1√x dx =
1 Why is it 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.
∞ is not a number: use t, then let t → ∞
2 Use a limit
The integral is the limit of the integral from 1 to t, as t goes to infinity.
∫ from 1 to ∞ of 1√x dx = lim t → ∞ ∫ from 1 to t of 1√x dx
3 Integrate
The power rule: x to the minus 1/2 becomes x to the 1/2, divided by 1/2.
= lim t → ∞ [ 2√x ] from 1 to t
4 Let t go
Put in t, then take away the same with 1.
= lim t → ∞ (2√t − (2))
As t grows, 2 √t grows without end.
2√t → ∞
5 Decide
The limit is infinity: the area has no end.
The integral diverges.
the limit = ∞
Careful!
The curve goes down to 0, but too slowly: the area still grows for ever.
Careful: not the curve → 0, so the area is a number ✗
6 Check
Let's check with a calculator: up to t = 1000 the integral is about 61.25.
t = 1000: 61.25
Up to a million, about 1998.
It keeps growing.
Right!
t = 1 000 000: 1998
Let's look back.
First: why is it improper?.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Last: check.
Answer Careful: not the curve → 0, so the area is a number ✗
▶ Watch it solved
The question
Question: find the integral from 1 to infinity of 1 over (x^(3/2)).
∫ from 1 to ∞ of 1x^(3/2) dx =
1 Why is it 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.
∞ is not a number: use t, then let t → ∞
2 Use a limit
The integral is the limit of the integral from 1 to t, as t goes to infinity.
∫ from 1 to ∞ of 1x^(3/2) dx = lim t → ∞ ∫ from 1 to t of 1x^(3/2) dx
3 Integrate
The power rule: x to the minus 3/2 becomes x to the −1/2, divided by −1/2.
= lim t → ∞ [ −2√x ] from 1 to t
4 Let t go
Put in t, then take away the same with 1.
= lim t → ∞ (−2√t − (−2))
As t grows, minus 2 over √t goes to 0.
−2√t → 0
The t part goes to 0.
= 0 − (−2)
5 Decide
The limit is a number: the integral converges to 2.
An area that goes on for ever, but adds up to 2!
= 2
Careful!
Never put infinity in like a number.
We let t grow and watch where the answer goes.
Careful: not put in ∞: −1∞ ✗
6 Check
Let's check with a calculator: the integral up to t = 1000 is about 1.9368.
Already close to 2.
Right!
t = 1000: 1.9368
Let's look back.
First: why is it improper?.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Last: check.
Answer Careful: not put in ∞: −1∞ ✗
▶ Watch it solved
