In the three-dimensional space ℝ3, consider the hyperboloid H: x^2+y^2−z^2=1 and the ellipsoid E: 3x^2+3y^2+z^2=4.
(a) Describe explicitly the intersection C of H and E.
(b) Let P be a point of C. Find cosθ where θ is the acute angle of intersection of the hyperboloid H and the ellipsoid E at P.
(c) Let S be the solid lying inside H and E. Show that S is symmetric with respect to the xy-plane.
(d) Compute the volume V of S (remember to simplify the result). Indication: express V as the difference of two integrals.
(e) Compute the surface area of S
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the three-dimensional space ℝ3, consider the hyperboloid H: x^2+y^2−z^2=1 and the ellipsoid E: 3x^2+3y^2+z^2=4.
(a) Describe explicitly the intersection C of H and E.
(b) Let P be a point of C. Find cosθ where θ is the acute angle of intersection of the hyperboloid H and the ellipsoid E at P.
(c) Let S be the solid lying inside H and E. Show that S is symmetric with respect to the xy-plane.
(d) Compute the volume V of S (remember to simplify the result). Indication: express V as the difference of two integrals.
(e) Compute the surface area of S
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