I'm looking to model the heat transfer by conduction from the aperture to the base of a compound parabolic dish (CPE). Here's what I have so far, thanks to CFDFEAGURU's solution for a cone at https://www.physicsforums.com/threads/heat-transfer-by-conduction-in-a-truncated-cone.368972/ let θ = the acceptance angle; let a' = the radius of the base; let a = the radius of the aperture, where a=a'/sin(θ); let L = the height of the CPE from base to aperture, where L=(a+a')/tan(θ); let T1 be the temperature of a'; let T2 be the temperature of a; where T2>T1; y(Φ)=((2a'(1+sin(θ)sin(Φ-θ))/(1-cos(Φ))-a, gives us the radius at Φ; z(Φ)=((2a'(1+sin(θ)cos(Φ-θ))/(1-cos(Φ)), gives us the height at Φ; Solving z for Φ gives us, Φ(z)=2cot-1((sec(θ)sqrt(zcos(θ)a(sin(θ)+1)+a(sin(θ)+1)2)-tan(θ)a(sin(θ)+1))/a(sin(θ)+1)); A(x)=π•(y(Φ(x)))2; Q'•dx/A(x)=-k*dt; Integrate the left side from 0 to L. Integrate the right side from T1 to T2. The calculus gets very hairy at the function composition (y ° Φ)(x). It's even too dense for Wolfram Alpha, which times out, given such a monstrosity. It looks to me like a differential equation, but that looks almost as hairy. Is there a better integral technique to address this mess?