Question

How do you find the area using the trapezoidal approximation method, given `(x^2-x)dx`, on the interval [0,2] with n=4?

Answer 1

The interval `[0,2]` is split into `4` equal intervals of `1//2`. Pictured are the four functions, the intervals, and the areas created by making trapezoids from the function points.

The first three intervals are actually triangles since they have `0`-points that rest on the `x`-axis.

Using the points

• `f(0)=0`
• `f(1/2)=-1/4`
• `f(1)=0`
• `f(3/2)=3/4`
• `f(2)=2`

we can calculate all the needed areas. The area for a trapezoid is given by `A=1/2h(b_1+b_2)`. Areas below the `x`-axis are negative.

`A_"total"=-1/2(1/2)(1/4)-1/2(1/2)(1/4)+1/2(1/2)(3/4)+1/2(1/2)(3/4+2)`

`=3/4`