To estimate the area under the curve $f(x) = 5 + x^2$ on the interval $[-2, 4]$ using three rectangles and right endpoints, follow these steps:

1. Divide the Interval:

The interval $[-2, 4]$ has a length of $4 - (-2) = 6$. Dividing it into 3 equal subintervals gives: $\text{Width of each subinterval} = \frac{6}{3} = 2$ The subintervals are: [ [-2, 0], [0, 2], [2, 4] ]

2. Use Right Endpoints:

The right endpoints for the subintervals are: $x_1 = 0, \quad x_2 = 2, \quad x_3 = 4$

3. Calculate $f(x)$ at Right Endpoints:

Using $f(x) = 5 + x^2$: $f(x_1) = f(0) = 5 + 0^2 = 5$ $f(x_2) = f(2) = 5 + 2^2 = 5 + 4 = 9$ $f(x_3) = f(4) = 5 + 4^2 = 5 + 16 = 21$

4. Calculate the Area of Each Rectangle:

The area of each rectangle is given by: $\text{Area} = (\text{width of interval}) \times f(\text{right endpoint})$ $\text{Area of Rectangle 1} = 2 \times f(x_1) = 2 \times 5 = 10$ $\text{Area of Rectangle 2} = 2 \times f(x_2) = 2 \times 9 = 18$ $\text{Area of Rectangle 3} = 2 \times f(x_3) = 2 \times 21 = 42$

5. Add the Areas:

The total estimated area under the curve is: $\text{Total Area} = 10 + 18 + 42 = 70$

Final Answer:

The estimated area under the curve is: $\boxed{70}$

Let me know if you'd like a graph of the setup or further clarification. Here are five related questions for further exploration:

1. What happens if we use left endpoints instead of right endpoints for the estimate?
2. How does the accuracy change with more subintervals?
3. Can you calculate the exact area using definite integration for comparison?
4. How would using the midpoint of each subinterval change the estimate?
5. How can the Trapezoidal Rule refine the estimate?

Tip: To improve accuracy in estimating areas under curves, increase the number of subintervals or use advanced techniques like Simpson's Rule!