Use a Riemann sum with 4 rectangles of equal width to approximate the area between $y=x^{3}+2$ and the $x$ axis on the interval $[-1,1]$. Use the right-hand endpoint of each subinterval.
9 units $^{2}$
3.5 units $^{2}$
8.0625 units $^{2}$
4.5 units $^{2}$

Step 1 ：We are given the function \(y=x^{3}+2\) and the interval \([-1,1]\). We are asked to approximate the area between the curve and the x-axis using a Riemann sum with 4 rectangles of equal width. The right-hand endpoint of each subinterval will be used.

Step 2 ：The width of each rectangle (dx) is the total width of the interval divided by the number of rectangles. In this case, \(dx = \frac{b - a}{n} = \frac{1 - (-1)}{4} = 0.5\).

Step 3 ：The right-hand endpoints of the subintervals are \(-0.5, 0, 0.5, 1\).

Step 4 ：The height of each rectangle is the value of the function at the right-hand endpoint of the subinterval. So, the heights are \(1.875, 2, 2.125, 3\).

Step 5 ：The area of each rectangle is the width times the height. So, the areas of the rectangles are \(0.5 \times 1.875, 0.5 \times 2, 0.5 \times 2.125, 0.5 \times 3\).

Step 6 ：The total area is the sum of the areas of the rectangles. So, the total area is \(0.5 \times 1.875 + 0.5 \times 2 + 0.5 \times 2.125 + 0.5 \times 3 = 4.5\) units \(^{2}\).

Step 7 ：Final Answer: The approximate area between \(y=x^{3}+2\) and the \(x\) axis on the interval \([-1,1]\) using a Riemann sum with 4 rectangles of equal width and the right-hand endpoint of each subinterval is \(\boxed{4.5}\) units \(^{2}\).