Use finite approximations to estimate the area under the graph of the function ​f(x)equals3 minus x squared minus 2 x between xequalsnegative 3 and xequals1 for each of the following cases.
a. Using a lower sum with two rectangles of equal width
b. Using a lower sum with four rectangles of equal width
c. Using an upper sum with two rectangles of equal width
d. Using an upper sum with four rectangles of equal width
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Part 1
a. The area under the graph of​ f(x) using a lower sum with two rectangles of equal width is nbsp
  
enter your response here. ​(Type an integer or a​ decimal.)

Let's break down the problem and solve each part step by step.

### Given Function:
\[
f(x) = 3 - x^2 - 2x
\]
### Interval:
\[
x \in [-3, 1]
\]

The width of each rectangle for a lower or upper sum is determined by dividing the interval into equal parts. Then, we'll calculate the function values at the appropriate points.

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### Part (a): Lower sum with 2 rectangles of equal width

1. **Step 1: Calculate the width of each rectangle**
   - Total width of the interval = \(1 - (-3) = 4\)
   - Width of each rectangle = \( \frac{4}{2} = 2\)

2. **Step 2: Subdivide the interval**
   - Subintervals: \([-3, -1]\) and \([-1, 1]\)

3. **Step 3: Determine the minimum value of \(f(x)\) on each subinterval**
   - For \([-3, -1]\): Evaluate \(f(-3)\) and \(f(-1)\)
     \[
     f(-3) = 3 - (-3)^2 - 2(-3) = 3 - 9 + 6 = 0
     \]
     \[
     f(-1) = 3 - (-1)^2 - 2(-1) = 3 - 1 + 2 = 4
     \]
     Minimum value: \(f(-3) = 0\)

   - For \([-1, 1]\): Evaluate \(f(-1)\) and \(f(1)\)
     \[
     f(1) = 3 - (1)^2 - 2(1) = 3 - 1 - 2 = 0
     \]
     Minimum value: \(f(1) = 0\)

4. **Step 4: Calculate the area of the lower sum**
   - Area = \((\text{width}) \times (\text{minimum value})\) for each rectangle:
     \[
     \text{Area} = (2)(0) + (2)(0) = 0
     \]

### Answer for Part (a): 
\[
\text{Lower sum with 2 rectangles} = 0
\]

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Would you like me to proceed with the other parts (b, c, and d) or provide a detailed explanation of part (a) first?

Use finite approximations to estimate the area under the graph of the function f(x) = 3 - x^2 - 2x between x = -3 and x = 1 for each of the following cases:

a. Using a lower sum with two rectangles of equal width
b. Using a lower sum with four rectangles of equal width
c. Using an upper sum with two rectangles of equal width
d. Using an upper sum with four rectangles of equal width

Explore how to estimate the area under the graph of f(x) = 3 - x^2 - 2x using finite approximations. This guide covers lower and upper sums with 2 and 4 rectangles over the interval [-3, 1], including detailed step-by-step calculations for each case.

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