Title 78 Surface Area and Length of Arc.pdf ✅

MSTC 78: Surface Area and Length of Arc 1. Length of Arc 1.1. Cartesian Coordinates [DERIVATION] From the Pythagorean Theorem, dL = √(dx) 2 + (dy) 2 Or dL = √1 + ( dy dx) 2 dx = √1 + ( dx dy) 2 dy Which results in the equation for the lengths of the arc L = ∫ √1 + ( dy dx) 2 dx x2 x1 L = ∫ √1 + ( dx dy) 2 dy y2 y1
1.2. Polar Coordinates [DERIVATION] From the Pythagorean Theorem, dL = √(r dθ) 2 + (dr) 2 or dL = √r 2 + ( dr dθ) 2 dθ Therefore, the formula is L = ∫ √r 2 + ( dr dθ) 2 dθ θ2 θ1 1.3. Parametric Equations [DERIVATION] From the cartesian coordinates, dL = √(dx) 2 + (dy) 2 = √( dx dt) 2 + ( dy dt) 2 dt It results in the formula L = ∫ √( dx dt) 2 + ( dy dt) t 2 2 t1 dt
2. Surface Area The formulas below come from the First Proposition of Pappus, A = 2πRL 2.1. Cartesian Coordinates The formulas are the following: SA = 2π ∫ y√1 + ( dy dx) 2 dx x2 x1 SA = 2π ∫ x√1 + ( dx dy) 2 dy y2 y1 2.2. Polar Coordinates The formula is based on the axis of revolution. 2.2.1. Revolved about the x-axis SA = 2π ∫ r sin θ √1 + ( dr dθ) 2 dθ θ2 θ1 2.2.2. Revolved about the y-axis SA = 2π ∫ r cos θ √1 + ( dr dθ) 2 dθ θ2 θ1