Applications of Integration

turning ∫ into area & volume 📦

📐 area between curves

Between y = f(x) and y = g(x) on [a,b] (assuming f ≥ g): A = ∫_a^b [f(x) − g(x)] dx

For x as a function of y, integrate horizontally with ∫ [right − left] dy.

Revolve y = R(x) about the x-axis: thin disks of radius R(x), thickness dx.

V = π ∫_a^b [R(x)]² dx

Hollow solids: outer radius R(x), inner r(x).

V = π ∫_a^b ([R(x)]² − [r(x)]²) dx

Same idea revolving around other axes/lines — measure radii perpendicular to the axis.

About vertical axis often: V = 2π ∫ radius · height · (thickness along parallel variable)

Pick dx vs dy shells so each strip is parallel to axis of revolution (classic "coffee can" heuristic).

L = ∫_a^b √(1 + [f′(x)]²) dx

For parametrics x(t), y(t): L = ∫ √((dx/dt)² + (dy/dt)²) dt. Polar r(θ): L = ∫ √(r² + (dr/dθ)²) dθ.

Area bounded by y = x² and y = √x.
Volume when y = √x from x = 0 to 4 is revolved around the x-axis.
Set up (don't evaluate badly) ∫ for arc length of y = ln(cos x) over an interval.

parametric & polar party next 🎢