Edexcel C4 Chapter 6 H section question 3 part b. Numerical integration

The first part of this question asks me to approximate the value of $\displaystyle\int^1_0 e^x tanx\ dx$

first with two strips, then with four, and finally with eight.


I do my thing and obtain the answers without difficulty.



The second part then asks me to suggest 'a possible value for I'

Because I don't know the exact value of $\displaystyle\int^1_0 e^x tanx\ dx$ I offer the approx for eight strips---this, it turns out, is wrong.

I go to the solutions at the end of the book (C4 edexcel in this case) to see the answer only to be confronted with an answer I am not sure I quite like or agree with at all.

It says that when you halve the widths of the trapeziums that the 'differences' decrease by a third. In a table below, the three approximations for each case (2 strips, 4 strips, etc) are displayed:
1.5098
1.329
1.282
Below and between these figures are displayed these so-called 'differences' (IMPORTANT see attachment below)

Going from 0.18 to 0.05 clearly IS NOT a reduction of 1/3

Then--to make matters even more confusing-- the solution suggests a forecast difference of 0.01/2 which makes no sense at all to me.

Can you explain their reasoning please? What does the word 'differences' mean in this context--Is it the result of subtracting 1.329 from 1.5098?, or do they mean the gaps left under/over the function by the trapezoids? It seems to me--at least in this case that either:

I have to calculate the exact value of $\displaystyle\int^1_0 e^x tanx\ dx$ using integration by parts (which in this case I don't seem to know how to do)

or take another approximation with a larger n

Have a look at the original document