Revise: Applications of Integration

∫ₐᵇ f(x)dx = F(b) − F(a), for any antiderivative F of f.

∫₁³2x dx = 9−1 = 8.

Area = ∫ₐᵇ f(x)dx, when f(x)≥0 on [a,b].

∫₁²x²dx = 7/3 ≈ 2.333.

Area = ∫ₐᵇ[f(x)−g(x)]dx, top minus bottom.

x+2 vs x² on [−1,2]: area=4.5.

V = π∫ₐᵇ[f(x)]²dx — one curve, rotated about the axis.

√x on [0,4] rotated: V=8π.

V = π∫ₐᵇ([R(x)]²−[r(x)]²)dx — two curves, a hole through the solid.

x vs x² on [0,1] rotated: V=2π/15.

Always evaluate at the upper limit first, then subtract the lower.

Reversing the order flips the sign.

The foundation for both area-between-curves and volumes of revolution.

A negative f(x) needs the interval split and absolute values taken.

Find intersection points first — they are usually the integration bounds.

Test a point inside the interval to confirm which curve is on top.

Each disk has area π[f(x)]² and thickness dx.

Only valid when the region touches the axis directly.

Square each radius separately — never square their difference.

[R(x)]²−[r(x)]², not (R(x)−r(x))².