Consider the function f(x) = 7 + x^2. Estimate the area under f from x = -1 to x = 2 using the methods below.
Six rectangles and mid points, entered into a variable M_6.

To estimate the area under the curve \( f(x) = 7 + x^2 \) from \( x = -1 \) to \( x = 2 \) using six rectangles and the midpoints method, we follow these steps:

### 1. Partition the Interval
The interval \([-1, 2]\) has a length of:
\[
\Delta x = \frac{2 - (-1)}{6} = \frac{3}{6} = 0.5
\]
Thus, the width of each rectangle is \( \Delta x = 0.5 \).

### 2. Midpoints
We will now compute the midpoints of each subinterval. For six rectangles, the subintervals are:
- \([-1, -0.5]\)
- \([-0.5, 0]\)
- \([0, 0.5]\)
- \([0.5, 1]\)
- \([1, 1.5]\)
- \([1.5, 2]\)

The midpoints for these intervals are:
\[
\text{Midpoints: } x_1 = -0.75, x_2 = -0.25, x_3 = 0.25, x_4 = 0.75, x_5 = 1.25, x_6 = 1.75
\]

### 3. Function Values at Midpoints
Now, calculate the function values \( f(x) \) at these midpoints:
\[
f(x) = 7 + x^2
\]
\[
f(-0.75) = 7 + (-0.75)^2 = 7 + 0.5625 = 7.5625
\]
\[
f(-0.25) = 7 + (-0.25)^2 = 7 + 0.0625 = 7.0625
\]
\[
f(0.25) = 7 + (0.25)^2 = 7 + 0.0625 = 7.0625
\]
\[
f(0.75) = 7 + (0.75)^2 = 7 + 0.5625 = 7.5625
\]
\[
f(1.25) = 7 + (1.25)^2 = 7 + 1.5625 = 8.5625
\]
\[
f(1.75) = 7 + (1.75)^2 = 7 + 3.0625 = 10.0625
\]

### 4. Estimation of Area
Now, multiply each function value by the width \( \Delta x = 0.5 \) and sum them up to get the estimated area:
\[
M_6 = 0.5 \times [f(-0.75) + f(-0.25) + f(0.25) + f(0.75) + f(1.25) + f(1.75)]
\]
\[
M_6 = 0.5 \times [7.5625 + 7.0625 + 7.0625 + 7.5625 + 8.5625 + 10.0625]
\]
\[
M_6 = 0.5 \times [47.875]
\]
\[
M_6 = 23.9375
\]

Thus, the estimated area under the curve using six rectangles and the midpoint rule is \( M_6 \approx 23.9375 \).

Would you like further clarification on any steps, or more details?

### Related Questions:
1. How does increasing the number of rectangles affect the accuracy of the midpoint rule?
2. How is the midpoint rule different from the trapezoidal rule for estimating areas?
3. Can the exact area under \( f(x) = 7 + x^2 \) from \( x = -1 \) to \( x = 2 \) be computed using integration?
4. What would the result be if we used the left or right Riemann sum instead of the midpoint rule?
5. How does the choice of partition points (midpoints vs. endpoints) influence the approximation?

### Tip:
For better approximations of areas under curves, increase the number of rectangles or use more advanced methods like Simpson’s rule.

Consider the function f(x) = 7 + x^2. Estimate the area under f from x = -1 to x = 2 using six rectangles and midpoints, entered into a variable M_6.

Learn how to estimate the area under the curve f(x) = 7 + x^2 from x = -1 to x = 2 using the midpoint rule and six rectangles. This step-by-step guide covers integral approximation, partitioning intervals, and calculating function values at midpoints. Suitable for high school and early college students.

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