Volume from Riemann Sums

Review: Area via Riemann Sums

• Target: area under y=f(x) on [a,b]
• Replace curved region with n rectangles (width \Delta x, height f(x_i))
• Approximation: \sum{i=1}^{n} f(xi)\,\Delta x
• Limit n\to\infty ⇒ exact area: \int_a^b f(x)\,dx

Extending the Idea to Volume

• Goal: volume of a 3-D solid (pyramid, cylinder, irregular shape)
• Strategy parallels 2-D case: replace solid with many easy solids (rectangular prisms, cylinders, etc.)

Volume Approximation Process

• Partition solid into n simple solids with easily computed volumes V_i
• Approximate volume: \sum{i=1}^{n} Vi
• Refine partition (more, smaller pieces) ⇒ better fit
• Limit n\to\infty: \lim{n\to\infty} \sum{i=1}^{n} V_i = \text{Volume of solid}

From Sums to Integrals

• The limiting sum becomes a volume integral (Riemann integral in 3-D)
• Symbolically: \int_{\text{solid}} dV or reduced to \int \text{(cross-sectional area)}\,dx depending on symmetry
• Same first-principles framework: “approximate → sum → limit → integral” now applied in three dimensions