Module 7 Quick Reference

Applications of Integration – Calculus I

Area Between Curves

Vertical strips (dx)

A = ∫ab [top − bottom] dx

Horizontal strips (dy)

A = ∫cd [right − left] dy

Find intersections first! If curves cross, split the integral.

Disk / Washer Method

Disk: V = π ∫ R² dx  |  Washer: V = π ∫ (R² − r²) dx

Slices

Perpendicular to axis of rotation

Non-standard axis

Measure R, r as distance from curve to axis

Shell Method

V = 2π ∫ (radius)(height) dx (or dy)

About y-axis

radius = x, height = f(x), integrate dx

About x-axis

radius = y, height = g(y), integrate dy

Shells: slices parallel to axis | Disks: slices perpendicular to axis

Average Value

favg = (1/(b − a)) ∫ab f(x) dx

MVT for Integrals

There exists c in [a, b] with f(c) = favg.

Geometric: rectangle of height favg has same area as region under f.

Decision Guide: Disk/Washer vs. Shell

Rotate about x-axis + y = f(x) easy → Disk/Washer (dx)

Rotate about y-axis + y = f(x) easy → Shell (dx)

Multiple integrals needed? → Try the other method