1. ## estimate integral

Estimate

ln 1.5 = (integral from 1 to 1.5) dt/t

using the approximation (1/2)[Lf(P) + Uf(P)] with
P = {1 = (4/4) , (5/4), (6/4) , (7/4) , (8/4), (9/4), (10/4) = (5/2)} .

sorry everyone! the book is online so i had copied and pasted this question but for some reason the numbers changed and i didnt notice. im sorry for the confusion. thank you all for letting me know.
[note- the f's in Lf and Uf are like L(sub)f and U(sub)f]

2. Originally Posted by runner07
P = {1 = 8 , 9 , 10 , 11 , 12 = 1. 5}.
1) I cannot imagine what that might mean.
2) If you invented this notation, please don't invent new notation. There is already way too much.
3) Can you provide the EXACT wording of the question and some indication of your efforts? If your textbook's author invented the notation, the explanation has to be in there, somewhere.

3. Originally Posted by runner07
Estimate

ln 1.5 = (integral from 1 to 1.5) dt/t

using the approximation (1/2)[Lf(P) + Uf(P)] with
P = {1 = 8 , 9 , 10 , 11 , 12 = 1. 5}.
I believe he is describing a Darboux integral

P describes the partitions and L(f,P) is the lower Darboux sum and U(f,P) is the upper Darboux sum

4. That may be, but I'm still wondering how 1 = 8 and 12 = 1.5!

Perhaps this means the number of subintervals?

5. Originally Posted by TKHunny
That may be, but I'm still wondering how 1 = 8 and 12 = 1.5!

Perhaps this means the number of subintervals?
yeah, he lost me there too

6. Originally Posted by runner07
Estimate

ln 1.5 = (integral from 1 to 1.5) dt/t

using the approximation (1/2)[Lf(P) + Uf(P)] with
P = {1 = 8 , 9 , 10 , 11 , 12 = 1. 5}.

Say the partition is $P=\{1,1.1,1.2,1.3,1.4,1.5\}$.

Then, $U(f,P) = \sup_{[1,1.1]}\{ f\}\cdot (1.1-1) + \sup_{[1.1,1.2]}\cdot (1.2-1.1) + ... + \sup_{[1.4,1.5]}\cdot (1.5-1.4)$
Since $f$ is non-increasing the supremum is the left endpoint.
Thus,
$U(f,P) = f(1)(.1)+f(1.1)(.1)+...+f(1.4)(.1) = .1 \left( 1+\frac{1}{1.1}+...+\frac{1}{1.4} \right)$
Similarly,
$L(f,P) = f(1.1)(.1)+f(1.2)(.1)+...+f(1.5)(.1) = .1\left( \frac{1}{1.1}+\frac{1}{1.2}+...+\frac{1}{1.5} \right)$

7. sorry everyone! the book is online so i had copied and pasted this question but for some reason the numbers changed and i didnt notice. im sorry for the confusion. thank you all for letting me know.

8. No worries. Use TPH's very clear demonstration and let's see what you get.