The parametric equation of the eliptic paraboloid: $${r}(u,v) = \left(u\cos(v); u\sin(v); \dfrac{1}{4}u^2-\dfrac{21}{4}\right)\qquad 0 \leq v \leq 2\pi,\,\,0 \leq u \leq +\infty$$ The parametric equation of the cone: $${r}(u,v) = \left(u\cos(v); u\sin(v); -u\right)\qquad 0 \leq v \leq 2\pi,\,\,0 \leq u \leq +\infty $$ The parametric equation of the cylinder: $${r}(u,v) = \left(\cos(v)+1; \sin(v); u\right)\qquad 0 \leq v \leq 2\pi,\,\,-\infty \leq u \leq +\infty $$
I found that,
• The parametric equation of the intersection curve of the cylinder and the cone: $$\left(\cos(t)+1,\sin(t),\dfrac{1}{2}\cos(t)-\dfrac{19}{4}\right),\qquad 0\leq t\leq 2\pi$$ • The parametric equation of the intersection curve of the cylinder and the eliptic paraboloid: $$\left(\cos(t) + 1,\sin(t),-\sqrt{2 + 2cos(t)}\right),\qquad 0\leq t\leq 2\pi$$
But how to define the area of the cylinder limited by two curves above?