Compute an approximation to $$\int_1^2\frac{1}{x}\,dx\,,$$ which gives the area under $y=\frac{1}{x}$ for $1\le x\le 2$, using a modified Riemann sum with the (NOT equally spaced) partition $$1,1.1,1.1^2,1.1^3,1.1^4,1.1^5,1.1^6,1.1^7,2$$ and left hand endpoints EXCEPT neglecting the area of the last rectangle. This amounts to computing the sum of the areas of the rectangles as shown in the following figure:

As you can see in the figure, the area of the last rectangle is relatively small compared to the others, and the other rectangles already give an overestimate of the area.

Please note that the problem is NOT asking for the value of $\int_1^2\frac{1}{x}\,dx$. Rather it is asking for the EXACT values of the areas of the 7 approximating rectangles and for the EXACT value of the sum of the areas of the rectangles. Calculator approximations (no matter how accurate) will NOT be accepted. Do the calculations by hand using fractions (until you notice the pattern in the areas).

The area of the first rectangle =
The area of the second rectangle =
The area of the third rectangle =
The area of the fourth rectangle =
The area of the fifth rectangle =
The area of the sixth rectangle =
The area of the seventh rectangle =
The sum of the areas of the 7 rectangles =