1. **Stating the problem:** We are given the integral $$\pi \int_{-2}^2 (4 - x^2) \, dx$$ and asked to identify which solid's volume it represents. 2. **Understanding the integral:** The integral $$\int_{-2}^2 (4 - x^2) \, dx$$ represents the area under the curve $$y = 4 - x^2$$ from $$x = -2$$ to $$x = 2$$. 3. **Volume by revolution:** Multiplying this integral by $$\pi$$ suggests the volume of a solid obtained by revolving the curve $$y = 4 - x^2$$ around the x-axis between $$x = -2$$ and $$x = 2$$. 4. **Formula for volume of revolution:** The volume $$V$$ of a solid formed by revolving $$y = f(x)$$ about the x-axis from $$a$$ to $$b$$ is $$V = \pi \int_a^b [f(x)]^2 \, dx$$. 5. **Check the integrand:** Here, the integrand is $$4 - x^2$$, not squared. So the integral $$\pi \int_{-2}^2 (4 - x^2) \, dx$$ is not directly the volume by revolution formula. 6. **Calculate the integral:** $$\int_{-2}^2 (4 - x^2) \, dx = \left[4x - \frac{x^3}{3}\right]_{-2}^2 = \left(4(2) - \frac{2^3}{3}\right) - \left(4(-2) - \frac{(-2)^3}{3}\right) = \left(8 - \frac{8}{3}\right) - \left(-8 + \frac{-8}{3}\right) = \left(8 - \frac{8}{3}\right) + 8 - \frac{8}{3} = 16 - \frac{16}{3} = \frac{48}{3} - \frac{16}{3} = \frac{32}{3}$$ 7. **Multiply by $$\pi$$:** $$\pi \times \frac{32}{3} = \frac{32\pi}{3}$$ 8. **Compare with known volumes:** - Volume of a sphere: $$\frac{4}{3} \pi r^3$$ - Volume of a right circular cone: $$\frac{1}{3} \pi r^2 h$$ - Volume of a right circular cylinder: $$\pi r^2 h$$ 9. **Check each option:** - (a) Sphere radius 4: $$\frac{4}{3} \pi 4^3 = \frac{4}{3} \pi 64 = \frac{256\pi}{3}$$ (not equal) - (b) Cone height 4, radius 3: $$\frac{1}{3} \pi 3^2 4 = \frac{1}{3} \pi 9 \times 4 = 12\pi$$ (not equal) - (c) Sphere radius 2: $$\frac{4}{3} \pi 2^3 = \frac{4}{3} \pi 8 = \frac{32\pi}{3}$$ (matches) - (d) Cylinder height 4, radius 2: $$\pi 2^2 4 = 16\pi$$ (not equal) 10. **Conclusion:** The volume $$\pi \int_{-2}^2 (4 - x^2) \, dx = \frac{32\pi}{3}$$ matches the volume of a sphere with radius 2 units. **Final answer:** (c) a sphere whose radius length is 2 units.