We are tasked with estimating the area under the graph of the function $f(x) = 7 + 4x^2$ between $x = -1$ and $x = 2$, using two methods: right-endpoint and midpoint approximations with three rectangles.

1. Using Right Endpoints

We divide the interval $[-1, 2]$ into three equal subintervals. The total width of the interval is:

$\Delta x = \frac{2 - (-1)}{3} = 1$

The right endpoints for these subintervals are:

• First subinterval: $[-1, 0]$, right endpoint is $x_1 = 0$
• Second subinterval: $[0, 1]$, right endpoint is $x_2 = 1$
• Third subinterval: $[1, 2]$, right endpoint is $x_3 = 2$

Now, the function values at the right endpoints are:

• $f(0) = 7 + 4(0)^2 = 7$
• $f(1) = 7 + 4(1)^2 = 7 + 4 = 11$
• $f(2) = 7 + 4(2)^2 = 7 + 16 = 23$

The area estimate using right endpoints is:

$R_3 = \Delta x \times (f(0) + f(1) + f(2)) = 1 \times (7 + 11 + 23) = 41 \, \text{square units}$

2. Using Midpoints

We use midpoints of each subinterval for this estimation. The midpoints for the subintervals are:

• First subinterval $[-1, 0]$, midpoint is $x_1 = -0.5$
• Second subinterval $[0, 1]$, midpoint is $x_2 = 0.5$
• Third subinterval $[1, 2]$, midpoint is $x_3 = 1.5$

Now, the function values at the midpoints are:

• $f(-0.5) = 7 + 4(-0.5)^2 = 7 + 4(0.25) = 8$
• $f(0.5) = 7 + 4(0.5)^2 = 7 + 4(0.25) = 8$
• $f(1.5) = 7 + 4(1.5)^2 = 7 + 4(2.25) = 16$

The area estimate using midpoints is:

$M_3 = \Delta x \times (f(-0.5) + f(0.5) + f(1.5)) = 1 \times (8 + 8 + 16) = 32 \, \text{square units}$

Summary:

• $R_3 = 41$ square units (Right-endpoints)
• $M_3 = 32$ square units (Midpoints)

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the results change if we used four rectangles instead of three?
2. How do left-endpoint approximations compare to right-endpoint approximations?
3. What is the exact area under the curve if we compute the definite integral of $f(x)$?
4. Can you estimate the error between the right-endpoint approximation and the exact integral?
5. How does the midpoint approximation improve accuracy compared to the right-endpoint approximation?

Tip: When estimating areas using Riemann sums, using more rectangles generally leads to better approximations.