To approximate the area beneath the curve using right endpoints and four subintervals on $[1,9]$, follow these steps:

Step 1: Determine the width of each subinterval.

The interval $[1,9]$ is divided into 4 subintervals. The total width is: $\Delta x = \frac{9 - 1}{4} = 2$ Each subinterval is 2 units wide: $[1,3], [3,5], [5,7], [7,9]$.

Step 2: Identify the right endpoints.

The right endpoints for each subinterval are:

• $[1,3]$: right endpoint = $x=3$
• $[3,5]$: right endpoint = $x=5$
• $[5,7]$: right endpoint = $x=7$
• $[7,9]$: right endpoint = $x=9$

Step 3: Find $f(x)$ at the right endpoints.

From the table, the function values at the right endpoints are:

• $f(3) = 4$
• $f(5) = 6$
• $f(7) = 3$
• $f(9) = 3$

Step 4: Apply the Riemann sum formula.

The area approximation using right endpoints is: $\text{Area} = \Delta x \cdot [f(3) + f(5) + f(7) + f(9)]$ Substitute the values: $\text{Area} = 2 \cdot [4 + 6 + 3 + 3] = 2 \cdot 16 = 32$

Final Answer:

$\boxed{32}$

Let me know if you want further clarification or additional details! Here are some related questions for deeper understanding:

1. How would the approximation change if left endpoints were used instead of right endpoints?
2. What is the role of $\Delta x$ in this calculation?
3. How does increasing the number of subintervals affect the approximation?
4. How would you approximate the area using the midpoint rule?
5. Can this problem be solved analytically if $f(x)$ were given as a continuous function?

Tip: Always double-check whether endpoints are inclusive and what type of rule is specified (right, left, or midpoint).