Originally Posted by

**HallsofIvy** There is no 'finding' $\displaystyle (i- 1/n)\Delta x$, that is an arbitrary **choice**. Once you have divided the interval 0 to 1 in n subintervals, each having length $\displaystyle \Delta x$, and so right endpoints $\displaystyle \Delta x$, $\displaystyle 2\Delta x$, $\displaystyle 3\Delta x$, ..., $\displaystyle i\Delta x$, ..., you must choose one x in each interval. Here they have just chosen the simplest- the midpoint: $\displaystyle i\Delta x+ (1/2)\Delta x= (i+ 1/2)\Delta x$. You could as easily have chosen $\displaystyle (i+ 1/4)\Delta x$ or $\displaystyle (i+ 1/3)\Delta x$ or $\displaystyle (i+3/4)\Delta x$, etc.

In the last problem, all of the individual areas are rectangles or trapezoids. Do you know the formulas for area of a rectangle or trapezoid? Find the area of each and add them.