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

Step 1: Determine the width of each rectangle

The interval is $[-2, 4]$, and it is divided into 3 subintervals. The width of each subinterval is: $\Delta x = \frac{4 - (-2)}{3} = \frac{6}{3} = 2.$

Step 2: Identify the right endpoints

The right endpoints for the three subintervals are:

1. Subinterval $[-2, 0]$: Right endpoint = $x = 0$
2. Subinterval $[0, 2]$: Right endpoint = $x = 2$
3. Subinterval $[2, 4]$: Right endpoint = $x = 4$

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

The function $f(x) = 5 + x^2$. Plugging in the right endpoints:

1. $f(0) = 5 + (0)^2 = 5$
2. $f(2) = 5 + (2)^2 = 5 + 4 = 9$
3. $f(4) = 5 + (4)^2 = 5 + 16 = 21$

Step 4: Calculate the total area

The area of each rectangle is given by: $\text{Area of rectangle} = (\text{Height}) \times (\Delta x).$ So, the total area is: $\text{Total Area} = \Delta x \left[f(0) + f(2) + f(4)\right].$ Substituting the values: $\text{Total Area} = 2 \left[5 + 9 + 21\right] = 2 \times 35 = 70.$

Final Answer:

The estimated area under the curve is 70.

Would you like further details or have any specific questions?

Related Questions:

1. How would the solution change if left endpoints were used instead of right endpoints?
2. How does the choice of the number of rectangles affect the accuracy of the estimation?
3. What is the exact value of the integral for this function on the interval $[-2, 4]$?
4. How does this approach relate to the Riemann Sum definition of an integral?
5. Can we apply midpoint rectangles for a more accurate estimate? How?

Tip:

For better accuracy in approximations, increase the number of rectangles ($n$) to reduce the width of each subinterval.