AP Calculus BC · Unit 8 Applications of Integration
Learning Objectives
Evaluate surface area for y=f(x) revolved around x-axis
Understand ds as arc length element
Recognize 2πr·ds as frustum surface area
Generalize to parametric x(t), y(t)
Formulas
Arc Length ds:
ds = √[1 + (f'(x))²] dx
Surface Area (x-axis):
S = ∫ 2π·f(x)·ds
S = ∫ 2π·f(x)·√[1 + f'(x)²] dx
Parametric form:
S = ∫ 2π·y(t)·√[x'(t)² + y'(t)²] dt
Tags
surface arearevolutionarc length
CURVE FUNCTION f(x)
Rotation Angle θ 2π
Upper Bound b 1.0
Calculated Surface Area
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Calculated Volume (Ref)
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Watch the curve generated around the X-axis. Note that Area requires the arc length term `ds = √(1+(dy/dx)²)` while Volume only requires `πy²dx` rings.