# hmwk7 - intersect along a diameter at an angle of π 6...

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Math 215 Homework Set 7: §§ 16.6 – 16.8 Fall 2007 Most of the following problems are modiﬁed versions of the recommended homework problems from your text book Multivariable Calculus by James Stewart. 16.6a. Find the region E for which the triple integral ± ± ± E (6 - 3 x 2 - 2 y 2 - 2 z 2 ) dV is a maximum. 16.6b. Find the center of mass of the tetrahedron bounded by the planes x = 0 , y = 0 , z = 0 , x +3 y +2 z = 6 ; ρ ( x, y, z ) = z . 16.6c. Sketch the region of integration for the integral ± 3 0 ± 9 9 - x 2 ± 9 - y 0 f ( x, y, z ) dz dy dx. Rewrite this integral as an equivalent iterated integral in three of the ﬁve possible other orders. 16.6d. Find the center of mass of the cube given by - a x a , - a y a and 0 z 2 a ; ρ ( x, y, z ) = x 2 + y 2 + z 2 . 16.6e. Do Problems 33 and 34 of § 16.6 in Stewart’s Multivariable Calculus . 16.8a. Find the volume of one of the smaller wedges cut from a sphere of radius 27 by two planes that
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Unformatted text preview: intersect along a diameter at an angle of π/ 6 . 16.8b. Find the volume and center of mass of the solid that lies above the cone z = 4 ² x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = 16 . Assume that the density of the solid is constant. 16.8c. Find the volume of the solid that lies above the cone ϕ = π/ 6 and below the sphere ρ = 9 cos( ϕ ) . 16.8d. Evaluate ± ± ± B (3 x 2 + 3 y 2 + 3 z 2 ) dV where B is the ball of radius 13 centered at the origin. 16.8e. Find the center of mass of a solid hemisphere of radius 5 if the density at any point in the hemisphere is proportional to the point’s distance from the base of the hemisphere. 16.8f. Do Problems 27–28 of § 16.7 39 and 40 of § 16.8 in Stewart’s Multivariable Calculus ....
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