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 3.65625 .
(Type an integer or a simplified fraction.)
Using four rectangles to estimate, the area under $f(x)$ is approximately
(Type an integer or a simplified fraction.)
Answer
Final Answer: Using two rectangles to estimate, the area under \(f(x)\) is approximately \(\boxed{3.65625}\). Using four rectangles to estimate, the area under \(f(x)\) is approximately \(\boxed{3.7265625}\).
Step 1 ：The problem is asking to estimate the area under the curve of the function \(f(x) = x^3\) between \(x=1\) and \(x=2\) using the midpoint rule with two and four rectangles.
Step 2 ：The midpoint rule is a numerical method for approximating the definite integral of a function. The interval [a, b] is divided into n subintervals, and the height of each rectangle is the value of the function at the midpoint of that rectangle's base. The area of each rectangle is then the base times the height, and the sum of these areas is the approximation of the integral.
Step 3 ：For two rectangles, the interval [1, 2] is divided into two subintervals: [1, 1.5] and [1.5, 2]. The midpoints of these intervals are 1.25 and 1.75, respectively. The height of each rectangle is then \(f(1.25)\) and \(f(1.75)\), respectively.
Step 4 ：For four rectangles, the interval [1, 2] is divided into four subintervals: [1, 1.25], [1.25, 1.5], [1.5, 1.75], and [1.75, 2]. The midpoints of these intervals are 1.125, 1.375, 1.625, and 1.875, respectively. The height of each rectangle is then \(f(1.125)\), \(f(1.375)\), \(f(1.625)\), and \(f(1.875)\), respectively.
Step 5 ：The areas of the two rectangles are 0.9765625 and 2.6796875, respectively. The total area under the curve estimated using two rectangles is the sum of these areas, which is approximately \(3.65625\).
Step 6 ：The areas of the four rectangles are 0.35595703125, 0.64990234375, 1.07275390625, and 1.64794921875, respectively. The total area under the curve estimated using four rectangles is the sum of these areas, which is approximately \(3.7265625\).
Step 7 ：Final Answer: Using two rectangles to estimate, the area under \(f(x)\) is approximately \(\boxed{3.65625}\). Using four rectangles to estimate, the area under \(f(x)\) is approximately \(\boxed{3.7265625}\).
