The table below shows some values of the increasing function *f* ( *x* ).

- Use a right-hand sum with one sub-interval to estimate the area between the graph of
*f* and the *x*-axis on the interval [2,8].

- Use a right-hand sum with three sub-intervals to estimate the area between the graph of
*f* and the *x*-axis on the interval [2,8].

- Are your answers in (a) and (b) over- or under-estimates of the actual area between the graph of
*f* and the *x*-axis on the interval [2,8]?

Answer

- If we use only one sub-interval, that means we're not breaking up the original interval up at all. We will have one rectangle of width 6 and height

*f* (8) = 20.

The area of this rectangle is

height ⋅ width = 20(6) = 120.

Since this is the only rectangle we're using, we estimate that the area between the graph of *f* and the *x*-axis is 120.

- To use three sub-intervals we need to break the original interval [2,8] into sub-intervals of length 2.On sub-interval [2,4] the height of the rectangle is
*f* (4) = 5 so the area of this rectangle is

height ⋅ width = 5(2) = 10.

On sub-interval [4,6] the height of the rectangle is *f* (6) = 12 so the area of this rectangle is

height ⋅ width = 12(2) = 24.

On sub-interval [6,8] the height of the rectangle is *f* (8) = 20 so the area of this rectangle is

height ⋅ width = 20(2) = 40.

Adding the areas of all the rectangles, we estimate that the area between the graph of *f* and the *x*-axis is

10 + 24 + 40 = 74.

- The estimates in (a) and (b) are over-estimates.

Since the function *f* is increasing, it must look something like this:

The rectangles must be covering extra area that isn't between the graph of *f* and the *x* axis: