Question

How do I estimate the area under the graph of `f(x)=25-x^2` from x=0 to x=5 using five rectangles and the right-endpoint method?

Answer 1

Estimate of area (using 5 right-endpoint rectangles = `70`

Explanation:

Using `5` rectangles (of width `1` each)
the right end point will have a height of
`color(white)("XXXX")` `f(1) = 24` for the rectangle from `0` to `1`
`color(white)("XXXX")` `f(2) = 21` for the rectangle from `1` to `2`
`color(white)("XXXX")` `f(3) = 16` for the rectangle from `2` to `3`
`color(white)("XXXX")` `f(4) = 9` for the rectangle from `3` to `4`
`color(white)("XXXX")` `f(5) = 0` for the rectangle from `3` to `5`

and since each rectangle has a width of 1
the corresponding rectangles will have areas:
`color(white)("XXXX")` `24, 21, 16, 9, 0`

The sum of the areas of the `5` rectangles gives an estimate of
the area under `f(x) = 25-x^2` from `0` to `5`

`24+21+16+9+0 = 70`