Question

How do you approximate the area under `y=10−x^2` on the interval [1, 3] using 4 subintervals and midpoints?

Answer 1

First you divide the interval in four equal parts, and in each part you find the midpoint.

These would be:
`[1,1.5]` gives `1.25` as a midpoint
`[1.5,2]` gives `1.75`
`[2,2.5]` gives `2.25`
`[2.5,3]` gives `2.75`

Then you plug in the values in the function (I will only do the first)
`y_1=10-1.25^2=8.4375`
Since each of the subintervals is `0.5` wide, the surface under the first quarter of the interval `[1,3]` can be approximated by:
`A_1=0.5*y_1=0.5*8.4375=4.21875`

You do the same for the other three parts and add. (Of course you round the total, as 5 decimals is a bit steep for an approxiation!)

Extra :
The exact area is `17 2/3` (calculated by using integrals).