Solution: The capsule consists of a cylinder of height $10 - 2 \times 2 = 6$ units (subtracting the hemispheres) and two hemispheres (equivalent to one sphere). Volume of the cylinder: $\pi r^2 h = \pi \times 2^2 \times 6 = 24\pi$. Volume of the sphere: $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi \times 8 = \frac{32}{3}\pi$. Total volume: $24\pi + \frac{32}{3}\pi = \frac{104}{3}\pi$. \boxed{\dfrac{104}{3}\pi}

["# Understanding the Volume of a Composite Geometric Shape: Cylinder and Hemispheres Combined", "When studying three-dimensional geometry, understanding how to calculate the volume of complex shapes is essential. One interesting shape combines a cylinder and two hemispheres—effectively, a cylinder flanked by attached hemispherical ends forming a complete sphere’s surface. In this article, we break down the volume calculation for such a composite solid and explain how to derive the total volume step by step.", "---", "## The Geometry: Cylinder with Attached Hemispheres", "Imagine a solid comprising two main parts:", "- A cylindrical section with height $ h = 6 $ units\n- Two hemispheres (each with radius $ r = 2 $ units) attached at the cylinder’s ends.", "When two hemispheres are joined end-to-end across a shared diameter, they form a full sphere—but geometrically, this setup preserves the cylinder’s height excluding the hemispherical "caps."", "---", "## Step 1: Calculate the Cylinder’s Volume", "The volume of any cylinder is given by the formula:", "[\nV_{\ ext{cylinder}} = \pi r^2 h\n]", "Here, the radius $ r = 2 $ and the effective height $ h = 6 $ units (since the hemispheres do not increase the vertical span of the solid). Plugging in the values:", "[\nV_{\ ext{cylinder}} = \pi \ imes 2^2 \ imes 6 = \pi \ imes 4 \ imes 6 = 24\pi\n]", "---", "## Step 2: Calculate the Volume of the Hemispheres", "Each hemisphere has volume:", "[\nV_{\ ext{hemisphere}} = \frac{2}{3} \pi r^3\n]", "But since two hemispheres form a complete sphere, their combined volume equals that of a full sphere:", "[\nV_{\ ext{two hemispheres}} = \frac{4}{3} \pi r^3\n]", "With $ r = 2 $:", "[\nV_{\ ext{sphere part}} = \frac{4}{3} \pi \ imes 2^3 = \frac{4}{3} \pi \ imes 8 = \frac{32}{3}\pi\n]", "---", "## Step 3: Total Volume of the Composite Solid", "Now sum both volumes:", "[\nV_{\ ext{total}} = V_{\ ext{cylinder}} + V_{\ ext{sphere}} = 24\pi + \frac{32}{3}\pi\n]", "To combine these terms, convert $24\pi$ to thirds:", "[\n24\pi = \frac{72}{3}\pi\n]", "Then add:", "[\nV_{\ ext{total}} = \frac{72}{3}\pi + \frac{32}{3}\pi = \frac{104}{3}\pi\n]", "---", "## Final Result", "[\n\boxed{\dfrac{104}{3}\pi}\n]", "---", "## Why This Matters", "Understanding volume calculations for composite shapes is vital in fields like engineering, architecture, and product design. This example shows how basic geometric formulas combine seamlessly to solve real-world spatial problems. Whether designing storage tanks, calculating material requirements, or modeling complex objects, mastering volume addition is a foundational skill.", "---", "Keywords: volume calculation, cylinder volume, center hemisphere cubes, geometric volume, composite shape volume, π radius cylinder, hemispherical caps, STEM education, engineering geometry."]