Use sigma notation to write the following Riemann sum. Then, evaluate the Riemann sum using formulas for the sums of powers of positive integers or a calculator.

The right Riemann sum for $f(x)=x+8$ on $[0,4]$ with $n=50$.

Step 1 ：The right Riemann sum is a method for approximating the total area underneath a curve on a graph, also known as an integral. The right Riemann sum approximates this area by dividing the area into rectangles and adding up the areas of these rectangles. The height of each rectangle is determined by the value of the function at the right endpoint of the subinterval.

Step 2 ：The formula for the right Riemann sum is given by: \(R_n = \sum_{i=1}^{n} f(x_i) \Delta x\), where \(f(x_i)\) is the value of the function at the right endpoint of the i-th subinterval, \(\Delta x\) is the width of each subinterval, and the sum is taken over all n subintervals.

Step 3 ：In this case, the function is \(f(x) = x + 8\), the interval is \([0,4]\), and there are \(n=50\) subintervals. So, the width of each subinterval is \(\Delta x = \frac{4-0}{50} = \frac{4}{50} = 0.08\).

Step 4 ：The right endpoint of the i-th subinterval is \(x_i = 0 + i \Delta x = 0 + i(0.08) = 0.08i\).

Step 5 ：So, the right Riemann sum is: \(R_{50} = \sum_{i=1}^{50} f(0.08i) \Delta x = \sum_{i=1}^{50} (0.08i + 8) \Delta x\).

Step 6 ：Now, we can write this sum in sigma notation and then evaluate it using formulas for the sums of powers of positive integers or a calculator.

Step 7 ：The right Riemann sum for the function \(f(x) = x + 8\) on the interval \([0,4]\) with \(n=50\) subintervals is approximately 40.16. This is the approximate area under the curve \(f(x) = x + 8\) on the interval \([0,4]\).

Step 8 ：Final Answer: The right Riemann sum for \(f(x)=x+8\) on \([0,4]\) with \(n=50\) is approximately \(\boxed{40.16}\).