To estimate the area under the curve $f(x) = x^2 + 5x + 3$ over the interval $[0, 2]$ using the right-endpoint approximation with $n = 2$ partitions, we follow these steps:

Step 1: Divide the interval into $n = 2$ equal subintervals.

The interval $[0, 2]$ is divided into 2 partitions, so the width of each subinterval is:

$\Delta x = \frac{2 - 0}{2} = 1$

This gives two subintervals: $[0, 1]$ and $[1, 2]$.

Step 2: Determine the right-endpoints of these subintervals.

For the right-endpoint approximation, we evaluate the function at the right endpoint of each subinterval.

• The right endpoint of the first subinterval $[0, 1]$ is $x = 1$.
• The right endpoint of the second subinterval $[1, 2]$ is $x = 2$.

Step 3: Evaluate the function $f(x) = x^2 + 5x + 3$ at these right-endpoints.

• At $x = 1$: $f(1) = 1^2 + 5(1) + 3 = 1 + 5 + 3 = 9$

• At $x = 2$: $f(2) = 2^2 + 5(2) + 3 = 4 + 10 + 3 = 17$

Step 4: Compute the area approximation using the right-endpoint rule.

The right-endpoint approximation formula for the area is:

$\text{Area} \approx \Delta x \left[ f(x_1) + f(x_2) \right]$ where $x_1 = 1$ and $x_2 = 2$ are the right-endpoints.

$\text{Area} \approx 1 \left[ 9 + 17 \right] = 1 \times 26 = 26$

Final Answer:

The area under the curve using the right-endpoint approximation with $n = 2$ partitions is approximately 26.

Would you like to dive deeper into this method or explore another related concept? Here are some follow-up questions to help guide you:

1. How does the area approximation change if we use $n = 4$ partitions instead?
2. What is the left-endpoint approximation for the same interval and function?
3. How does the accuracy of the approximation improve with more partitions?
4. How would you calculate the exact area under the curve using integration?
5. What would happen if the function were changed to $f(x) = x^3 + 2x + 5$?

Tip: Using smaller partitions generally gives a more accurate approximation of the area, and increasing the number of partitions reduces the error in the estimate.