pls slove the math step by step

Question

Sketch the graph and find the points on the cardioid r = a(1 +cosθ ) at which tangent lines are horizontal. Also, find its surface area by revolving about initial line.

Answer 1

The initial line is the x-axis

S = 2π∫₀^π y ds

ds = √(r²+(r')² dθ = √(a²(1+cosθ)² + (-asinθ)²)dθ = a√(2(1+cosθ))dθ

S = 2πa²∫₀^π (1+cosθ)sinθ√2 √(1+cosθ) dθ

let u = 1+cosθ, du = -sinθ

when θ= 0, u = 2

when θ= π, u=0

S = -2√2 πa²∫₂⁰u^3/2du = 2√2 πa²∫₀²u^3/2du = 2√2(2/5)a²πu^5/2|₀²

S = 4√2πa²2^5/2/5 = 32a²/5