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Home Multiple Integrals Extra Problems for Chapter 12  
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Extra Problems for Chapter 12
1 Compute the Riemann sum 2 Compute the Riemann sum 3 Evaluate 4 Evaluate 5 Evaluate 6 Evaluate 7 Find the volume of the solid over the region  1 ≤ x ≤ 1, 0 ≤ y ≤ 1  x^{2} and between the surfaces z = 0, z = 1  y. 8 Find the volume of the solid over the region x^{2} + y^{1} = 4 and between the surfaces z = 0 and z = y^{2} + x + 2. 9 Find the volume of the solid 1 ≤ x ≤ 2, 0 ≤ y ≤ in x, y/x ≤ z ≤ 1/x. 10 Find the volume of the solid x^{2} + y^{2} ≤ 1, x^{2}y^{3} ≤ z ≤ 1. 11 Find the volume of the solid bounded by the planes x = 1, x = y, z = x + y, z = x + 2, 12 Find the volume of the solid bounded by the cylinders x^{2} + y^{2} = 1, x^{2} + z^{2} = 1. 13 Find the mass, center of mass, and moment of inertia about the origin of the plane object 0 ≤ x ≤ π, 0 ≤ y ≤ sin x, ρ(x, y) = k. 14 Find the mass, center of mass, and moment of inertia about the origin of the plane object 0≤x≤l, x≤y≤l, ρ(x, y) = x^{2}y. 15 A circular disc filling the region x^{2} + y^{2} ≤ r^{2} has density ρ(x, y) = y^{2}. Find the mass, center of mass, and moment of inertia about the origin. 16 A semicircular object on the region has density ρ(x, y) = y. Find the work required to stand the object up on its flat side. 17 Using polar coordinates, find the volume of the solid x^{2} + y^{2} ≤ 9, y ≤ z ≤ x + 5. 18 Find the volume of the solid over the region 0 ≤ r ≤ 3 + cos θ between the plane z = 0 and the cone z = r. 19 Find the volume of the solid over the circle 0 ≤ r ≤ a between the plane z = 0 and the surface z = 1/r. 20 Find the mass and the moment of inertia about the origin of a semicircular object 0 ≤ r ≤ l, 0 ≤ θ ≤ 7π whose density is ρ(r, θ) = rθ. 21 A plane object covers the circle 0 ≤ r ≤ a and its density depends only on the distance r from the center, ρ(r, θ) = f(r). Show that the center of mass is at the origin. 22 Evaluate 23 Evaluate 24 Evaluate the triple integral 25 An object has constant density k in the region Find its center of mass and its moments of inertia about the coordinate axes. 26 Use cylindrical coordinates to evaluate ∫∫∫_{E} z dV, where E is the region inside the cylinder x^{2} + y^{2} = 1 which is above the plane z = 0 and within the sphere x^{2} + y^{2} + z^{2} = 9. 27 An object of constant density k has the shape of a parabolic bowl Find its center of mass and its moment of inertia about the zaxis. 28 Use spherical coordinates to evaluate the integral E is the spherical octant 29 A spherical shell a ≤ ρ ≤ b has density equal to the distance from the center. Find its mass and its moment of inertia about a diameter. 30 Prove that the double Riemann sum ∑∑_{D} f(x, y) dx dy is finite whenever f(x,y) is continuous, D is a closed region, and dx, dy are positive infinitesimals. 31 Suppose a plane object is symmetric about the xaxis, that is, it covers a region D of the form and has density ρ(x, y) = ρ(x, y). Prove that the center of mass is on the xaxis. 32 The moment of inertia about the xaxis of a point in the plane of mass m is I_{x} = my^{2}. Use the Infinite Sum Theorem to show that the moment of inertia about the xaxis of a plane object with density ρ(x, y) in the region D is I_{x} = ∫∫_{D} ρ(x, y)y^{2} dA. 33 The kinetic energy of a point of mass m moving at speed v is KE = ½mv^{2}. A rigid object of density ρ(x, y) in the plane region D is rotating about the origin with angular velocity ω (so a point at distance d from the origin has speed ωd). Use the Infinite Sum Theorem to show that the kinetic energy of the object is .34 Suppose a plane object is symmetric about the origin; that is, it fills a polar region 0 ≤ r ≤ g(θ), π ≤ d ≤ π, such that g(θ ± π) = g(θ), and its density has the property ρ(r, θ) = ρ(r, θ ± π). Show that the center of mass is at the origin. 35 Use the Infinite Sum Theorem to show that if D is a polar region of the form a ≤ r ≤ b, α(r) ≤ θ ≤ β(r), then


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