We divide the interval into two sub-intervals [0,2] and [2,4], each of width 2: On sub-interval [0,2] we go to the **midpoint** of [0,2], which is *x* = 1. The value of the function at this midpoint gives the height of the rectangle, so the height of the rectangle is *f* (1) = 2: The area of this rectangle is height ⋅ width = 2(2) = 4. On sub-interval [2,4] we go to the **midpoint** of [2,4], which is *x* = 3. The value of the function at this midpoint gives the height of the rectangle, so the height of the rectangle is *f* (3) = 10: The area of this rectangle is height ⋅ width = 10(2) = 20. We estimate that the area between *f* and the *x*-axis on [0,4] is 4 + 20 = 24. |