Question

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

Answer 1

We are approximating an area from `a` to `b` with `a=0` and `b=5`, `n=5`, right endpoints and `f(x)=25-x^2`
(For comparison, we'll do the same problem, but use left endpoints after we finish this.)

We need `Delta x=(b-a)/n`

`Deltax` is both the base of each rectangle and the distance between the endpoints.

For this problems `Deltax=(5-0)/5=1`.

Now, find the endpoints. (All of them to start with.)

The left-most endpoint is `a`, which, in this problem, is `0`. Start adding `Deltax` until we get to the end of the Intevral we are interested in.

Endpoints: `a=0`,
`a+Deltax=0+1=1`,
the next endpoint is the previous endpoint plus `Deltax`, `1+Delta x= 1+1=2`,
then `2+1=3`, and so on 4,`and`5#.

The endpoints are: `0,1,2,3,4,5`.
The right endpoints are `1,2,3,4,5`

The heights at these endpoints are:
`f(1)=24`
`f(2)=21`
`f(3)=16`,
`f(4)=9` and
`f(5)=0`

The areas of the rectagles are `Deltax` times the heights.

`1*24=24`,
`1*21=21`,
`1*16=16` and so on.

The area can be approximated by adding the areas of the five rectangles:
`(1*24)+(1*21)+(1*16)+(1*9)+(1*0) =70`

We did not use the graph of the function, but here it is, if you want to look at it.

graph{25-x^2 [-4.72, 46.6, -1.03, 24.65]}

By way of comparison: Using the LEFT endpoints and `5` rectangles would have given us:
The LEFT endpoints are `0, 1,2,3,4,`

The heights at these left endpoints are:
`f(0)=25`
`f(1)=24`,
`f(2)=21`,
`f(3)=16`, and
`f(4)=9`

The areas of the rectagles are `Deltax` times the heights.

`1*25=25`,
`1*24=24`,
`1*21=21` and so on.

The area can be approximated by adding the areas of the five rectangles:
`(1*25)+(1*24)+(1*21)+(1*16)+(1*9)=95`.