 # 1. Find the mass of a star that is a ball of radius 3, centered at...

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1. Find the mass of a star that is a ball of radius 3, centered at the origin, and density g(x,y,z) = 10-x2-32-22. 2. Find the volume of a cone of height one and radius 1, bounded by the surface 2 = Vx2+y2 and the plane 1. 3. + where V is the region bounded by 2 = r2 + y2 and = 8 (x²+y²). 4. The equation a²z² = h2x2 + h-y2 is a cone with a point at the origin that opens upwards (and down), such that at height z = h, the radius of the circle is a. Find the volume inside this cone of height h and radius a at that height. 5. Find the volumes of two conical bodies, one bounded by the surface 2 = a y2 and 2 = 0, and the other bounded by the surfaces 2 = Vx+y2 and 2 = a. where a is a constant. Try to solve this problem by doing each cone separately and then by calculating just one triple integral. 7. Compute the volume integral of f(r,A,2 = pi3 over the region bounded by the paraboloid r2 = 9 and the plane z= 0. 1. If F = 4.xzî y2 yzk evaluate fs F.nds where S is the surface of the cube bounded by x=0, x=1, y=0, y=1, z=0, z=1. 2. IfF = yi+ (x - 2xz)² xyk evaluate SS,(VXF) .nds where S is the surface of the sphere 22 +y++2 = a2 above the xy plane. 3. Let 0 = 45.x2y and let V denote the closed region bounded by the planes 4.x + 2y + 2 = 8, x 0, y =0, = = 0. Evaluate odV. 4. evaluate FdV where Vis the region bounded by the surfaces x= 0,y= 0,y=6,z=r²,z=4. 5. Find the volume of the region common to the intersecting cylinders x2 -y2 = a² and x2 +2² = a2. 6. Verify the divergence theorem for A = lri-2y23 2² over the region bounded by x2+y2 = 4. z 0 and 3.

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