Approximate the area under the curve $y=x^{3}$ from $x=1$ to $x=3$ using a Right Endpoint approximation with 4 subdivisions.

Step 1 ：We are asked to approximate the area under the curve \(y=x^{3}\) from \(x=1\) to \(x=3\) using a Right Endpoint approximation with 4 subdivisions.

Step 2 ：The area under a curve can be approximated using the Riemann sum. In this case, we are using the Right Endpoint approximation. This means that for each subdivision, we will use the value of the function at the right endpoint of the subdivision to determine the height of the rectangle. The width of each rectangle will be the width of the subdivision. The area of each rectangle will be the height times the width, and the sum of the areas of all the rectangles will be our approximation of the area under the curve.

Step 3 ：The width of each subdivision can be calculated as \((b - a) / n\), where \(b\) is the upper limit of the interval, \(a\) is the lower limit, and \(n\) is the number of subdivisions. In this case, \(b = 3\), \(a = 1\), and \(n = 4\), so the width of each subdivision is \((3 - 1) / 4 = 0.5\).

Step 4 ：The right endpoint of each subdivision will be \(a + i * w\), where \(i\) is the index of the subdivision (starting from 1), and \(w\) is the width of each subdivision. The height of each rectangle will be the value of the function at the right endpoint, which is \((a + i * w)^{3}\).

Step 5 ：The area of each rectangle will be the height times the width, which is \((a + i * w)^{3} * w\). The sum of the areas of all the rectangles will be our approximation of the area under the curve.

Step 6 ：The final approximate area under the curve \(y=x^{3}\) from \(x=1\) to \(x=3\) using a Right Endpoint approximation with 4 subdivisions is \(\boxed{27.0}\).