The approximation to ∫02x2+1dx∫02​*x*2+1*dx*, by using the rectangles OABC*OABC* and CDEF*CDEF* with vertices O(0,0),A(0,1),B(1,1),C(1,0)*O*(0,0),*A*(0,1),*B*(1,1),*C*(1,0) and D(1,2),E(2,2),F(2,0)*D*(1,2),*E*(2,2),*F*(2,0), is

To approximate the integral \(\int_0^2 (x^2 + 1) \, dx\) using the rectangles \(OABC\) and \(CDEF\) with the given vertices, we'll calculate the area of each rectangle and sum them up.

### 1. Rectangle \(OABC\):
- **Vertices:** \(O(0, 0)\), \(A(0, 1)\), \(B(1, 1)\), \(C(1, 0)\)
- **Width:** The width of the rectangle is the distance along the \(x\)-axis, which is \(1 - 0 = 1\).
- **Height:** The height of the rectangle is the value of the function at \(x = 0\), which is \(f(0) = 0^2 + 1 = 1\).

The area of rectangle \(OABC\) is:
\[
\text{Area of } OABC = \text{width} \times \text{height} = 1 \times 1 = 1
\]

### 2. Rectangle \(CDEF\):
- **Vertices:** \(C(1, 0)\), \(D(1, 2)\), \(E(2, 2)\), \(F(2, 0)\)
- **Width:** The width of the rectangle is the distance along the \(x\)-axis, which is \(2 - 1 = 1\).
- **Height:** The height of the rectangle is the value of the function at \(x = 1\), which is \(f(1) = 1^2 + 1 = 2\).

The area of rectangle \(CDEF\) is:
\[
\text{Area of } CDEF = \text{width} \times \text{height} = 1 \times 2 = 2
\]

### Total Approximate Area
The approximation to the integral is the sum of the areas of these two rectangles:
\[
\text{Approximation} = \text{Area of } OABC + \text{Area of } CDEF = 1 + 2 = 3
\]

Thus, the approximation to \(\int_0^2 (x^2 + 1) \, dx\) using the rectangles \(OABC\) and \(CDEF\) is **3**.

### Related Questions:
1. How do we determine the accuracy of the rectangle method for approximating an integral?
2. What would be the result if we used more rectangles of smaller width for the approximation?
3. How does the trapezoidal rule differ from the rectangle method for approximating integrals?
4. What are other numerical methods available for approximating definite integrals?
5. How would the result change if the function were \(x^3 + 1\) instead of \(x^2 + 1\)?

### Tip:
The rectangle method is a basic numerical integration technique, but increasing the number of rectangles (using smaller intervals) or using more advanced methods like Simpson's rule can provide more accurate results.

Learn how to approximate the integral of x^2 + 1 from 0 to 2 using the rectangular approximation method. This problem covers numerical integration techniques, specifically the rectangular method, with step-by-step calculations and results. Suitable for students in Grades 11-12.

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