For a volume formed by discs with outer radius R(x), which formula gives the volume over [a,b]?

For a volume formed by discs with outer radius R(x), which formula gives the volume over [a,b]?

• A. ∫_a^b (R)^2 dx
• B. π ∫_a^b (R)^2 dx
• C. π ∫_a^b R dx
• D. ∫_a^b (R^2) dx

Answer: (B)

This uses the disk method: each cross-sectional disk has area A(x) = π[R(x)]^2. A thin disk of thickness dx contributes π[R(x)]^2 dx to the volume. Integrating from a to b sums all those disks, giving V = ∫_a^b π[R(x)]^2 dx, which is the same as π ∫_a^b [R(x)]^2 dx. The π is essential because the area of a circle is π times the radius squared. Without the π or without squaring the radius, the expression wouldn’t represent volume.

This uses the disk method: each cross-sectional disk has area A(x) = π[R(x)]^2. A thin disk of thickness dx contributes π[R(x)]^2 dx to the volume. Integrating from a to b sums all those disks, giving V = ∫_a^b π[R(x)]^2 dx, which is the same as π ∫_a^b [R(x)]^2 dx. The π is essential because the area of a circle is π times the radius squared. Without the π or without squaring the radius, the expression wouldn’t represent volume.