Let f(x) = 7 + 4x2. Estimate the area under the graph of f, the x-axis, and the lines x = −1 and x = 2 using three rectangles and a right endpoints. Repeat part (a) using a midpoints. M3 = M6 =

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.

Let f(x) = 7 + 4x^2. Estimate the area under the graph of f, the x-axis, and the lines x = −1 and x = 2 using three rectangles and a right endpoints. Repeat part (a) using a midpoints. M3 = M6 =

Estimate the area under the curve of f(x) = 7 + 4x^2 using right-endpoint and midpoint Riemann sum approximations with three rectangles between x = -1 and x = 2. This problem introduces key concepts in area estimation and Riemann sums, ideal for students in Grades 11-12.

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