Use finite approximations to estimate the area under the graph of the function ​f(x)equals15 minus x squared plus 2 x between xequalsnegative 3 and xequals5 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
Question content area bottom
Part 1
a. The area under the graph of​ f(x) using a lower sum with two rectangles of equal width is nbsp
  
0. ​(Type an integer or a​ decimal.)
Part 2
b. The area under the graph of​ f(x) using a lower sum with four rectangles of equal width is nbsp
  
16. ​(Type an integer or a​ decimal.)

The function you're working with is:

\[
f(x) = 15 - x^2 + 2x
\]

The goal is to estimate the area under this curve between \(x = -3\) and \(x = 5\) using finite approximations (lower and upper sums). The length of the interval is \(5 - (-3) = 8\).

### Part 1a: Lower Sum with 2 Rectangles of Equal Width

1. **Determine the width of each rectangle:**

   Since the interval is \([-3, 5]\) and we're dividing it into 2 rectangles, the width of each rectangle is:

   \[
   \text{Width} = \frac{8}{2} = 4
   \]

2. **Find the left endpoints of each rectangle:**

   For a lower sum, we use the function values at the left endpoints.

   The left endpoints of the two rectangles are:
   - For the first rectangle: \(x_1 = -3\)
   - For the second rectangle: \(x_2 = 1\) (since \( -3 + 4 = 1\))

3. **Evaluate the function at the left endpoints:**

   - \(f(-3) = 15 - (-3)^2 + 2(-3) = 15 - 9 - 6 = 0\)
   - \(f(1) = 15 - 1^2 + 2(1) = 15 - 1 + 2 = 16\)

4. **Calculate the area using the lower sum formula:**

   \[
   \text{Area} = \text{width} \times [f(-3) + f(1)] = 4 \times [0 + 16] = 4 \times 16 = 64
   \]

   So, the area under the graph using a lower sum with 2 rectangles is **64**.

### Part 1b: Lower Sum with 4 Rectangles of Equal Width

1. **Determine the width of each rectangle:**

   The width of each rectangle is:

   \[
   \text{Width} = \frac{8}{4} = 2
   \]

2. **Find the left endpoints of each rectangle:**

   The left endpoints are:
   - \(x_1 = -3\)
   - \(x_2 = -1\) (since \( -3 + 2 = -1\))
   - \(x_3 = 1\)
   - \(x_4 = 3\)

3. **Evaluate the function at the left endpoints:**

   - \(f(-3) = 0\)
   - \(f(-1) = 15 - (-1)^2 + 2(-1) = 15 - 1 - 2 = 12\)
   - \(f(1) = 16\)
   - \(f(3) = 15 - 3^2 + 2(3) = 15 - 9 + 6 = 12\)

4. **Calculate the area using the lower sum formula:**

   \[
   \text{Area} = \text{width} \times [f(-3) + f(-1) + f(1) + f(3)] = 2 \times [0 + 12 + 16 + 12] = 2 \times 40 = 80
   \]

   So, the area under the graph using a lower sum with 4 rectangles is **80**.

### Part 1c: Upper Sum with 2 Rectangles of Equal Width

1. **Determine the width of each rectangle:**

   The width is still \(4\), as in Part 1a.

2. **Find the right endpoints of each rectangle:**

   For an upper sum, we use the function values at the right endpoints.

   The right endpoints of the two rectangles are:
   - For the first rectangle: \(x_1 = 1\)
   - For the second rectangle: \(x_2 = 5\)

3. **Evaluate the function at the right endpoints:**

   - \(f(1) = 16\)
   - \(f(5) = 15 - 5^2 + 2(5) = 15 - 25 + 10 = 0\)

4. **Calculate the area using the upper sum formula:**

   \[
   \text{Area} = \text{width} \times [f(1) + f(5)] = 4 \times [16 + 0] = 4 \times 16 = 64
   \]

   So, the area under the graph using an upper sum with 2 rectangles is **64**.

### Part 1d: Upper Sum with 4 Rectangles of Equal Width

1. **Determine the width of each rectangle:**

   The width is \(2\), as in Part 1b.

2. **Find the right endpoints of each rectangle:**

   The right endpoints are:
   -

Use finite approximations to estimate the area under the graph of the function f(x) = 15 - x^2 + 2x between x = -3 and x = 5 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.

Learn how to estimate the area under the curve of the function f(x) = 15 - x^2 + 2x using lower and upper sums. This problem involves dividing the interval into rectangles and approximating the area using finite sums. Suitable for students in Grades 11-12 and college-level calculus.

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