We have the function $f(x) = x^2$ on the interval $[0,2]$ and need to approximate the area under the curve using the Right Endpoint Rule with $n = 4$.

Step 1: Compute $\Delta x$

$\Delta x = \frac{b-a}{n} = \frac{2-0}{4} = \frac{2}{4} = 0.5$

Step 2: Identify Right Endpoints

The right endpoints are given by: $x_i = a + i \Delta x = 0 + i(0.5)$ for $i = 1,2,3,4$, giving: $x_1 = 0.5, \quad x_2 = 1.0, \quad x_3 = 1.5, \quad x_4 = 2.0$

Step 3: Compute Function Values

$f(0.5) = (0.5)^2 = 0.25$ $f(1.0) = (1.0)^2 = 1.00$ $f(1.5) = (1.5)^2 = 2.25$ $f(2.0) = (2.0)^2 = 4.00$

Step 4: Compute the Approximation

Using the Right Endpoint Rule: $R_n = \sum_{i=1}^{n} f(x_i) \Delta x$ $R_4 = (0.25 + 1.00 + 2.25 + 4.00) \times 0.5$ $R_4 = 7.5 \times 0.5 = 3.75$

Final Answer:

$R_4 = 3.75$

Would you like a more detailed breakdown or a graph of the rectangles?

Related Questions:

1. How would the left-endpoint approximation differ from the right-endpoint?
2. How does increasing $n$ affect the accuracy of the approximation?
3. Can we calculate the exact area using integration?
4. How does the midpoint rule compare to the right-endpoint rule for this function?
5. How would this process change if the function were $f(x) = x^3$?

Tip: The right endpoint rule overestimates the area for increasing functions and underestimates for decreasing functions.