Let's solve the problem **step by step**: ### **Step 1: Understand the Problem** - The football's **length** is 10.8 inches (from tip to tip). - The **diameter** at its thickest is 7 inches. - The football's shape is modeled by: \( f(x) = -0.12x^2 + 3.5 \) for \( -5.4 \leq x \leq 5.4 \) (since 10.8 / 2 = 5.4). ### **Step 2: Set Up the Volume Integral** When the region under \( f(x) \) is rotated about the x-axis, it forms a solid of revolution. The volume \( V \) is: \[ V = \pi \int_{a}^{b} [f(x)]^2 dx \] where \( a = -5.4 \), \( b = 5.4 \). ### **Step 3: Write the Integral** \[ V = \pi \int_{-5.4}^{5.4} \left(-0.12x^2 + 3.5\right)^2 dx \] ### **Step 4: Expand the Function** \[ \left(-0.12x^2 + 3.5\right)^2 = (-0.12x^2)^2 + 2(-0.12x^2)(3.5) + (3.5)^2 \] \[ = 0.0144x^4 - 0.84x^2 + 12.25 \] ### **Step 5: Plug Into the Integral** \[ V = \pi \int_{-5.4}^{5.4} \left(0.0144x^4 - 0.84x^2 + 12.25\right) dx \] ### **Step 6: Integrate Term by Term** - \(\int x^4 dx = \frac{x^5}{5}\) - \(\int x^2 dx = \frac{x^3}{3}\) - \(\int dx = x\) So: \[ V = \pi \left[ 0.0144 \cdot \frac{x^5}{5} - 0.84 \cdot \frac{x^3}{3} + 12.25x \right]_{-5.4}^{5.4} \] ### **Step 7: Plug in the Limits** Let’s compute each term at \( x = 5.4 \) and \( x = -5.4 \): #### Calculating at \( x = 5.4 \): - \( (5.4)^5 = 459.1656 \) - \( (5.4)^3 = 157.464 \) - \( 5.4 \) Plug in: - \( 0.0144 \cdot \frac{459.1656}{5} = 0.0144 \cdot 91.83312 = 1.3214 \) - \( 0.84 \cdot \frac{157.464}{3} = 0.84 \cdot 52.488 = 44.0899 \) - \( 12.25 \cdot 5.4 = 66.15 \) So, at \( x = 5.4 \): \[ 1.3214 - 44.0899 + 66.15 = 23.3815 \] At \( x = -5.4 \): - \( (-5.4)^5 = -459.1656 \) - \( (-5.4)^3 = -157.464 \) - \( -5.4 \) Plug in: - \( 0.0144 \cdot \frac{-459.1656}{5} = -1.3214 \) - \( 0.84 \cdot \frac{-157.464}{3} = -44.0899 \) - \( 12.25 \cdot (-5.4) = -66.15 \) So, at \( x = -5.4 \): \[ -1.3214 - (-44.0899) - 66.15 = -1.3214 + 44.0899 - 66.15 = -23.3815 \] ### **Step 8: Subtract to Get the Definite Integral** \[ V = \pi \left[ 23.3815 - (-23.3815) \right] = \pi (46.763) \] ### **Step 9: Calculate the Numerical Value** \[ V = \pi \times 46.763 \approx 3.1416 \times 46.763 \approx 146.86 \] ### **Final Answer** \[ \boxed{146.86} \text{ cubic inches (rounded to two decimal places)} \] --- **Summary:** - Set up the integral for volume of revolution - Expanded the function - Integrated term by term - Evaluated definite integral - Final answer: \(\boxed{146.86}\) cubic inches Let me know if you need further clarification or a breakdown of any step!