Let's break down and solve the problems step by step: --- ### **Step 1: Understanding the Bases and Cross-Sections** **Given functions and regions:** - \( f(x) = \sqrt{4-x^2} \): This is the upper semicircle of radius 2 centered at the origin. - \( g(x) = 2 + x \): This is a straight line. **Volumes:** - **A, B, C:** Use base bounded by \( y = f(x) \), \( y = 0 \) for \( x \in [-2, 2] \) - **D, E, F:** Use base bounded by \( y = g(x) \), \( y = 0 \), \( x = -2 \) to \( x = 2 \) **Cross-sections:** - A, D: quarter circles - B, E: squares - C, F: 45-45-90 right triangles --- ### **Step 2: Find Which Volumes Are Equal** The cross-sections for A/D, B/E, and C/F are the same, but the bases are different (\( f(x) \) vs \( g(x) \)). Let's check if the _dimensions of the base_ over the interval \([-2, 2]\) are the same for each pair. - For \( x \in [-2, 2] \): - \( f(x) \) is a semicircle (height varies from 0 to 2 and back to 0). - \( g(x) \) is a straight line from 0 to 4. **Volumes:** - **A, B, C** all have the same base (\( f(x) \)), but different cross-sections. - **D, E, F** all have the same base (\( g(x) \)), but different cross-sections. Let's compare the formulas: #### **For B and E (squares):** - B: \( V = \int_{-2}^{2} [f(x)]^2 dx \) - E: \( V = \int_{-2}^{2} [g(x)]^2 dx \) #### **For C and F (45-45-90 triangles):** - C: \( V = \int_{-2}^{2} \frac{1}{2}[f(x)]^2 dx \) - F: \( V = \int_{-2}^{2} \frac{1}{2}[g(x)]^2 dx \) #### **For A and D (quarter circles):** - A: \( V = \int_{-2}^{2} \frac{\pi}{4}[f(x)]^2 dx \) - D: \( V = \int_{-2}^{2} \frac{\pi}{4}[g(x)]^2 dx \) So, for each pair (A/D, B/E, C/F), the only difference is the function inside: \( f(x) \) vs. \( g(x) \). --- ### **Step 3: Are Any Volumes Equal?** - **Within each set (A/B/C or D/E/F)**: The cross-section shape is different, so the volumes are different. - **Across sets**: The cross-section type is the same, but the base is different (semicircle vs. line), so the volumes are not equal. Therefore, **none of the volumes are equal**. --- ### **Step 4: Answer Question 1** > **Which volumes are equal? If all are different, select only "A".** **Answer:** **A** (since all are different, per instructions) --- ### **Step 5: Rank the 3 Largest Volumes Left (Question 2)** We remove any equal volumes (there are none), so we rank all six volumes by size. But the question only asks for the 3 largest. Let's analyze the formulas: - For the same cross-section, **g(x) = 2 + x** over \([-2, 2]\) will have a larger area than \(f(x) = \sqrt{4-x^2}\) since \(g(x)\) is always above or equal to \(f(x)\) except at the endpoints. - For the same base, the order of cross-section areas is: 1. Quarter circle (\(\frac{\pi}{4} a^2\)) 2. Square (\(a^2\)) 3. Triangle (\(\frac{1}{2} a^2\)) Since \(\frac{\pi}{4} > 1 > \frac{1}{2}\), the quarter circle gives the largest volume for a given base. For \(g(x)\), the linear function, the area is greater than the semicircle except at the endpoints. So, the **three largest volumes** are for \(g(x)\) with the three cross-sections, ordered as: 1. Quarter circle (D) 2. Square (E) 3. Triangle (F) **So, in decreasing order:** **D** (quarter circle, g(x)) **E** (square, g(x)) **F** (triangle, g(x)) --- ### **Final Answers** #### **Question 1:** **A** #### **Question 2:** **D** **E** **F** --- **Explanation Summary:** - No volumes are equal because the bases are different. - For the same cross-section, \( g(x) \) provides a larger base than \( f(x) \). - For the same base, quarter circles have the largest area, then squares, then triangles. - The three largest are D, E, F in that order.

Let's analyze and compute each part step by step, providing detailed reasoning and calculations. --- ## **Part 1: Determining if volumes are equal** ### **Understanding the problem setup** - **Functions:** - \(f(x) = \sqrt{4 - x^2}\): semicircle of radius 2 centered at origin, \(x \in [-2, 2]\). - \(g(x) = 2 + x\): line from \(-2, 0\) to \(2, 4\). - **Bases:** - Regions bounded by \( y = f(x) \) and \( y = 0 \) for \(x \in [-2, 2]\). - Regions bounded by \( y = g(x) \) and \( y=0 \), with \(x \in [-2, 2]\). - **Cross-sections:** - For A, B, C: base is \( y=f(x) \). - For D, E, F: base is \( y=g(x) \). - **Cross-section shapes:** - A, D: quarter circles. - B, E: squares. - C, F: 45-45-90 right triangles. --- ### **Are any volumes equal?** Because the cross-sectional shapes differ (quarter circles vs squares vs triangles), and the bases are different functions, the corresponding volumes will differ. The only way for volumes to be equal is if both the base regions and the cross-sectional areas are identical, which they are not. **Conclusion:** **All volumes are different.** **Answer to Question 1:** **A** --- ## **Part 2: Calculating the volumes and ranking the 3 largest** ### **Step 1: Find the areas of the base regions** - **\(f(x) = \sqrt{4 - x^2}\):** semicircle of radius 2. - **\(g(x) = 2 + x\):** line from \(-2, 0\) to \(2, 4\). ### **Step 2: Cross-sectional areas** The volume is obtained by integrating the cross-sectional area along the base. --- ### **Step 3: Cross-sectional area formulas** - **Quarter circle (radius \(r\)):** area of a quarter circle of radius \(r\) is \(\frac{\pi r^2}{4}\). - **Square (side \(s\)):** area \(s^2\). - **45-45-90 triangle (legs \(a\)):** area \(\frac{1}{2} a^2\). --- ### **Step 4: Expressing the cross-sectional areas** **Note:** The cross-sectional areas at position \(x\) depend on the width of the base at \(x\): - **At \(x\), the height of the base:** - For \(f(x)\): height \(f(x) = \sqrt{4 - x^2}\). - For \(g(x)\): height \(g(x) = 2 + x\). --- ### **Step 5: Computing the volume formulas** For each shape, the volume is: \[ V = \int_{x_{start}}^{x_{end}} \text{(cross-sectional area)} \, dx \] --- ### **Part A: Volumes over base \(f(x)\)** - **A (quarter circle):** Cross-sectional area at \(x\): \[ A_{A}(x) = \frac{\pi}{4} \times [f(x)]^2 \] Since \( [f(x)]^2 = 4 - x^2 \), the volume: \[ V_A = \int_{-2}^{2} \frac{\pi}{4}(4 - x^2) dx \] Calculate: \[ V_A = \frac{\pi}{4} \int_{-2}^{2} (4 - x^2) dx \] - **B (square):** Cross-sectional area: \[ A_{B}(x) = [f(x)]^2 = 4 - x^2 \] Volume: \[ V_B = \int_{-2}^{2} (4 - x^2) dx \] - **C (triangle):** Cross-sectional area: \[ A_{C}(x) = \frac{1}{2} [f(x)]^2 = \frac{1}{2}(4 - x^2) \] Volume: \[ V_C = \int_{-2}^{2} \frac{1}{2} (4 - x^2) dx \] --- ### **Part D, E, F: over base \(g(x)\)** - **D (quarter circle):** \[ V_D = \int_{-2}^{2} \frac{\pi}{4} [g(x)]^2 dx \] - **E (square):** \[ V_E = \int_{-2}^{2} [g(x)]^2 dx \] - **F (triangle):** \[ V_F = \int_{-2}^{2} \frac{1}{2} [g(x)]^2 dx \] Where \(g(x) = 2 + x\). --- ### **Step 6: Compute the integrals** --- ### **Calculate integrals for \(f(x)\):** \[ \int_{-2}^{2} (4 - x^2) dx \] First, compute: \[ \int_{-2}^{2} 4 dx = 4 \times (2 - (-2)) = 4 \times 4 = 16 \] \[ \int_{-2}^{2} x^2 dx = \left[\frac{x^3}{3}\right]_{-2}^{2} = \frac{2^3}{3} - \frac{(-2)^3}{3} = \frac{8}{3} - \frac{-8}{3} = \frac{8}{3} + \frac{8}{3} = \frac{16}{3} \] Thus, \[ \int_{-2}^{2} (4 - x^2) dx = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3} \] --- ### **Volumes over \(f(x)\):** - **A:** \[ V_A = \frac{\pi}{4} \times \frac{32}{3} = \frac{\pi}{4} \times \frac{32}{3} = \frac{32 \pi}{12} = \frac{8 \pi}{3} \] - **B:** \[ V_B = \frac{32}{3} \] - **C:** \[ V_C = \frac{1}{2} \times \frac{32}{3} = \frac{16}{3} \] --- ### **Calculate integrals for \(g(x) = 2 + x\):** \[ \int_{-2}^{2} [g(x)]^2 dx = \int_{-2}^{2} (2 + x)^2 dx \] Expand: \[ (2 + x)^2 = 4 + 4x + x^2 \] Integrate term by term: \[ \int_{-2}^{2} 4 dx = 4 \times 4 = 16 \] \[ \int_{-2}^{2} 4x dx = 4 \times \left[\frac{x^2}{2}\right]_{-2}^{2} = 4 \times \left(\frac{4}{2} - \frac{4}{2}\right) = 4 \times (2 - 2) = 0 \] \[ \int_{-2}^{2} x^2 dx = \frac{16}{3} \quad (from earlier) \] Adding all: \[ \int_{-2}^{2} (4 + 4x + x^2) dx = 16 + 0 + \frac{16}{3} = 16 + \frac{16}{3} = \frac{48}{3} + \frac{16}{3} = \frac{64}{3} \] --- ### **Volumes over \(g(x)\):** - **D:** \[ V_D = \frac{\pi}{4} \times \frac{64}{3} = \frac{64 \pi}{12} = \frac{16 \pi}{3} \] - **E:** \[ V_E = \frac{64}{3} \] - **F:** \[ V_F = \frac{1}{2} \times \frac{64}{3} = \frac{32}{3} \] --- ## **Final Volume values:** | Volume | Expression | Numeric Approximate | |---------|--------------|---------------------| | \(V_A\) | \(\frac{8 \pi}{3}\) | \(\approx 8.377\) | | \(V_B\) | \(\frac{32}{3} \approx 10.667\) | | | \(V_C\) | \(\frac{16}{3} \approx 5.333\) | | | \(V_D\) | \(\frac{16 \pi}{3}\) | \(\approx 16.755\) | | \(V_E\) | \(\frac{64}{3} \approx 21.333\) | | | \(V_F\) | \(\frac{32}{3} \approx 10.667\) | | --- ## **Part 3: Rank the volumes from largest to smallest (excluding equal ones)** Order: 1. \(V_D \approx 16.755\) 2. \(V_E \approx 21.333\) 3. \(V_B \approx 10.667\) 4. \(V_F \approx 10.667\) 5. \(V_A \approx 8.377\) 6. \(V_C \approx 5.333\) **Largest three:** - 1st: **E** (\(\approx 21.333\)) - 2nd: **D** (\(\approx 16.755\)) - 3rd: **B** (\(\approx 10.667\)) --- ## **Final answers:** ### **Question 1:** **Are any volumes equal?** **No.** All are different. **Answer:** **A** ### **Question 2:** **Order of the three largest volumes:** \[ \boxed{ \text{1st: } E \\ \text{2nd: } D \\ \text{3rd: } B } \] --- ## **Summary:** | Volume | Name | Approximate value | |---------|--------|-------------------| | \(V_E\) | Square (g(x)) | 21.33 | | \(V_D\) | Quarter circle (g(x)) | 16.76 | | \(V_B\) | Square (f(x)) | 10.67 | | \(V_F\) | Triangle (g(x)) | 10.67 | | \(V_A\) | Quarter circle (f(x)) | 8.38 | | \(V_C\) | Triangle (f(x)) | 5.33 | --- **Note:** The calculations confirm the order and support the conclusion that all volumes are distinct, with the largest being the volume over base \(g(x)\) with square cross-sections (E). --- **If you have further questions or need clarification, feel free to ask!**