Question

How do you use the midpoint rule to approximate the integral `-3x-8x^2dx` from the interval [-1,4] with `n=3`?

Answer 1

Divide the interval `[-1,4]` into `3` equal width strips.
Find the mid point of each strip and evaluate `f(x)` at that point.
Multiply `f`(strip midpoint`) xx`strip-width# to get strip area.
Add the strip areas to get approximation of integral.

If `[-1,4]` is divided into `3` equal strips then each strip will be
`5/3` wide.

The strip mid points will be at
`-0.1667, 1.5,` and `3.1667`

`f(x) = - 3x - 8x^2`
`f(`midpoint of first strip`) = f(-0.1667) = 0.2778`
`f(`midpoint of second strip`) = f(1.5) = - 22.5`
`f(`midpoint of third strip`) = f(3.1667) = - 89.722`

Area of first strip `= 0.2778 xx 5/3 = 0.4630`
Area of second strip `= (- 22.5) xx 5/3 = -37.5`
Area of third strip #= (- 89.722) xx 5/3 = -149.537

Total area of strips
`0.4630 + (-37.5) + (-149.537)`
`= - 186.524`
which is our approximation for the integral