Question

How do you estimate the area under the graph of `y=2x^3+4x` from 0 to 3 using 4 approximating rectangles of equal width and right endpoints?

Answer 1

Dividing the width `0` to `3` into `4` rectangles of equal width
implies that each rectangle will have a width of `3/4`.
(This would have been so much easier with `3` rectangles).

The `1st` rectangle will have
Left Edge at `x=0` and Right Edge at `x=3/4`
with corresponding `y` values of Left y `= 0` and Right y `= 3.84`
(plugging the x Left and Right Edge values into the given formula to obtain the y Left and Right values).

...and so on for the `2nd`, `3rd`, and `4th` rectangles with corresponding Right y values of `12.75`, `31.78`, and `66`
(it's only the Right y values that of interest to us in this case).

The areas of the 4 rectangles (using the Right endpoints) are
`3/4 xx 3.84`,
`3/4 xx 12.75`,
`3/4 xx 31.78`, and
`3/4 xx 66`

Giving at total area of
`2.88 + 9.56 + 23.84 + 49.50`
`=85.78`

(Note that the graph is representational and not accurate, and that all calculations are only carried out to 2 significant places to the right of the decimal point).