Using rectangles whose height is given by the value of the function at the midpoint of the rectangle's base, estimate the area under the graph using first two and then four rectangles. \[ f(x)=x^{3} \text { between } x=1 \text { and } x=2 \] Using two rectangles to estimate, the area under $f(x)$ is approximately. (Type an integer or a simplified fraction.)

Step 1 ：We are given the function \(f(x) = x^3\) and we are asked to estimate the area under the curve between \(x=1\) and \(x=2\) using two rectangles and the midpoint rule.

Step 2 ：The midpoint rule approximates the area under the curve by dividing the area into rectangles and summing up the areas of these rectangles. The height of each rectangle is the value of the function at the midpoint of the base of the rectangle.

Step 3 ：For two rectangles, we need to divide the interval from 1 to 2 into two equal parts: [1, 1.5] and [1.5, 2]. The midpoints of these intervals are 1.25 and 1.75 respectively.

Step 4 ：We then evaluate the function at these midpoints and multiply by the width of the rectangles (which is 0.5) to get the area of each rectangle. The sum of these areas will give us the approximate area under the curve.

Step 5 ：Let's calculate this: midpoints = [1.25, 1.75], width = 0.5, area = 3.65625

Step 6 ：The area under the curve of the function \(f(x) = x^3\) between \(x=1\) and \(x=2\) is approximately 3.65625 when estimated using two rectangles and the midpoint rule. This is a reasonable approximation given the simplicity of the method used.

Step 7 ：Final Answer: The approximate area under the curve of the function \(f(x) = x^3\) between \(x=1\) and \(x=2\) using two rectangles and the midpoint rule is \(\boxed{3.65625}\).