Rucete ✏ AP Calculus In a Nutshell
7. Applications of Integration to Geometry
This chapter explores how definite integrals are used to compute areas, volumes, arc lengths, and more, especially in geometric contexts. Topics also include solids of revolution, polar areas, parametric arc lengths, and improper integrals.
Area Under a Curve
• For a nonnegative function f(x) on [a, b]: Area = ∫ab f(x) dx
• If f(x) changes sign: Break interval at zeros and integrate absolute values to find total area
• For x = g(y): Area = ∫ab g(y) dy
Area Between Curves
• If y = f(x) is above y = g(x) on [a, b]: Area = ∫ab [f(x) – g(x)] dx
• For intersections and changes in dominance, split into intervals
Using Symmetry
• Use symmetry to simplify area calculations:
– Symmetry about x-axis → double area above x-axis – Symmetry about y-axis → double area to the right – Symmetry about both axes → multiply area in first quadrant by 4
Polar Area
• Area enclosed by r = f(θ) from θ = α to θ = β:
A = (1/2) ∫αβ [f(θ)]² dθ
• Use symmetry and analyze intersection points carefully
Solids with Known Cross Sections
• If A(x) is the area of a cross section perpendicular to x-axis: Volume = ∫ab A(x) dx
• Common cross sections: – Squares: A(x) = [base]² – Semicircles: A(x) = (1/2)πr² – Equilateral triangles: A(x) = (√3 / 4)(base)²
Solids of Revolution (Disks and Washers)
Disks:
• Region rotated about axis with no hole: Volume = π ∫ab [r(x)]² dx
Washers:
• Region between two curves revolved: Volume = π ∫ab [R(x)]² – [r(x)]² dx
Shell Method
• Revolve vertical strip around y-axis (or horizontal around x-axis)
• Volume = 2π ∫ab (radius)(height) dx = 2π ∫ab x·f(x) dx (if around y-axis)
Examples of Solids
• Sphere from semicircle: Volume = (4/3)πr³
• Region under y = x² rotated about x- or y-axis (use washer or shell depending on axis)
Arc Length
• For y = f(x) on [a, b]: Arc length = ∫ab √[1 + (f′(x))²] dx
• For x = g(y) on [c, d]: Arc length = ∫cd √[1 + (g′(y))²] dy
Parametric Arc Length (BC)
• If x = f(t), y = g(t) on [a, b]:
Arc length = ∫ab √[(dx/dt)² + (dy/dt)²] dt
Surface Area of Revolution
• If a curve y = f(x) is revolved about the x-axis:
S = 2π ∫ab f(x) √[1 + (f′(x))²] dx
• If rotated about the y-axis: S = 2π ∫ab x √[1 + (f′(x))²] dx
Improper Integrals
• Type I: Infinite interval ∫a∞ f(x) dx = limt→∞ ∫at f(x) dx
• Type II: Infinite discontinuity ∫ab f(x) dx, where f is undefined at a, b, or in (a, b)
• Converges if the limit exists and is finite
Comparison Test for Improper Integrals
• If 0 ≤ f(x) ≤ g(x) for x ≥ a, and ∫g(x) converges, then ∫f(x) converges
• If ∫f(x) diverges and f(x) ≥ g(x) ≥ 0, then ∫g(x) diverges
In a Nutshell
Integrals are powerful tools for finding geometric quantities like area, volume, arc length, and surface area. Solids of revolution use disks, washers, and shells. Polar and parametric curves require special forms. Improper integrals extend integration to infinite limits and singularities using limits and convergence tests.