Volumes of RevolutionJEE Main

Concept

Spin the region under y = f(x) about the x-axis: every vertical strip becomes a disk of radius y and thickness dx, volume $\pi y^2 dx$. Sum → integrate: $V = \pi\int_a^b y^2\,dx$. Cones, spheres and every vase on a lathe fall out of one formula.

Key formula

Derivation

Riemann logic: n disks each of volume π·f(x_i)²·Δx; as n → ∞ the staircase of disks converges to the smooth solid.

Sphere check: y = √(r²−x²) on [−r, r]: $V = \pi\int(r^2 - x^2)dx = \pi[r^2x - x^3/3]_{-r}^{r} = \tfrac43\pi r^3$ ✓. Cone from y = (r/h)x: πr²h/3 ✓.

Scenarios to explore

• Volumes of Revolution — Disks stack into solids — π∫f² dx made visible.

Real-world applications

• Deriving the sphere/cone/paraboloid volume formulas honestly.
• Engineering: tank volumes, lathe-turned parts.
• Washer method for solids with holes (rotate the region between two curves).

JEE exam tips

• About y-axis via shells: V = 2π∫x·f(x)dx — sometimes far easier.
• Symmetric solids: integrate half, double.
• Sketch first — the axis of rotation determines radius = distance TO the axis.

Common mistakes

• Forgetting to SQUARE y.
• Rotating about y-axis with the x-axis formula (swap roles: V = π∫x²dy).
• Washer subtraction: π∫(R² − r²)dx, NOT π∫(R − r)²dx.

Exam traps to avoid

• Region between curve and axis BELOW the axis: y² handles the sign automatically.
• Rotating y = x² about y = −1: radius is (x² + 1), not x².