Jim Lambers MAT 280 Spring Semester Lecture 13 Notes

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1 Jim Lambers MAT 28 Spring Semester 29-1 Lecture 13 Notes These notes correspond to Sections 12.4 and 12.5 in Stewart and Sections 5.5 and 6.3 in Marsden and Tromba. Triple Integrals The integral of a function of three variables over a region R 3 can be defined in a similar way as the double integral. Let be the bo defined by {(, y, z) a b, c y d, r z s}. Then, as with the double integral, we divide [a, b] into n subintervals of width Δ (b a)/n, with endpoints [ i 1, i ], for i 1, 2,..., n. Similarly, we divide [c, d] into m subintervals of width Δy (d c)/m, with endpoints [y j 1, y j ], for j 1, 2,..., m, and divide [r, s] into l subintervals of width Δz (s r)/l, with endpoints [z k 1, z k ] for k 1, 2,..., l. Then, we can define the triple integral of a function f(, y, z) over by f(, y, z) dv lim m,n,l n m i1 j1 k1 l f( i, yj, zk ) ΔV, where ΔV ΔΔyΔz. As with double integrals, the practical method of evaluating a triple integral is as an iterated integral, such as f(, y, z) dv s d b r c a f(, y, z) d dy dz. By Fubini s Theorem, which generalizes to three dimensions or more, the order of integration can be rearranged when f is continuous on. A triple integral over a more general region can be defined in the same way as with double integrals. If is a bounded subset of R 3, that is contained within a bo B, then we can define f(, y, z) dv F (, y, z) dv, where F (, y, z) { f(, y, z) (, y, z), (, y, z) /. B 1

2 All of the properties previously associated with the double integral, such as linearity and additivity, generalize to the triple integral as well. Just as regions were classified as type I or type II for double integrals, they can be classified for the purpose of setting up triple integrals. A solid region is said to be of type 1 if it lies between the graphs of two continuous functions of and y that are defined on a two-dimensional region. Specifically, {(, y, z) (, y), u 1 (, y) z u 2 (, y)}. Then, an integral of a function f(, y, z) over can be evaluated as f(, y, z) dv u2 (,y) u 1 (,y) f(, y, z) dz da, where the double integral over can be evaluated in a manner that is appropriate for the type of. For eample, if is of type I, then and therefore {(, y, z) a b, g 1 () y g 2 (), u 1 (, y) z u 2 (, y)}, f(, y, z) dv b g2 () u2 (,y) a g 1 () u 1 (,y) f(, y, z) dz dy d. On the other hand, if is of type 2, then it has a definition of the form {(, y, z) (y, z), u 1 (y, z) u 2 (y, z)}. That is, lies between the graphs of two continuous functions of y and z that are defined on a two-dimensional region. Finally, if is a region of type 3, then it lies between the graphs of two continuous functions of and z. That is, {(, y, z) (, z), u 1 (y, z) y u 2 (y, z)}. If more than one type applies to a given region, then the order of evaluation can be determined by which ordering leads to the integrands that are most easily anti-differentiated within each single integral that arises. ample Let be a solid tetrahedron bounded by the planes, y, z and +y+z 1. We wish to integrate the function f(, y, z) z over this tetrahedron. From the given bounding planes, we see that the tetrahedron is bounded below by the plane z and above by the plane z 1 y. Therefore, we surmise that can be viewed as a solid of type 1. This requires finding a region in the y-plane such that is bounded by z and z 1 y on. 2

3 We first note that these planes intersect along the line + y 1. It follows that the base of is a 2- region that can be described by the inequalities, y, and + y 1. This region is of type I or type II, so we choose type I and obtain the description {(, y) 1, y 1 }. Therefore, we can integrate f(, y, z) over as follows: z dv y y ( z 2 1 y 2 z dz dy d z dz dy dz dy d (1 y) 2 dy d ) (1 y)3 1 d 3 (1 ) 3 d (1 u)u 3 du, u 1 1 u 3 u 4 du 6 1 ( ) u u ( ) ample We will compute the volume of the solid bounded by the surfaces y, y 2, z, and z. Because is bounded by two surfaces that define z as a function of and y, we view as a solid of type 1. It is bounded by the graphs of the functions z and z that are defined on a region in the y-plane. This region is bounded by the curves y and y 2. Because these curves intersect when and 1, we can describe as a region of type I: {(, y) 1, 2 y }. 3

4 It follows that the volume of is given by the iterated integral 1 dv 1 dz dy dz dy d 2 1 dy d ( 2 ) d 2 3 d ( ) Applications of ouble and Triple Integrals We now eplore various applications of double and triple integrals arising from physics. When an object has constant density ρ, then it is known that its mass m is equal to ρv, where V is its volume. Now, suppose that a flat plate, also known as a lamina, has a non-uniform density ρ(, y), for (, y), where defines the shape of the lamina. Then, its mass is given by m ρ(, y) da. Similarly, if is a solid region in 3- space, and ρ(, y, z) is the density of the solid at the point (, y, z), then the mass of the solid is given by m ρ(, y, z) dv. We see that just as the integral allows simple product formulas for area and volume to be applied to more general problems, it allows similar formulas for quantities such as mass to be generalized as well. The center of mass, also known as the center of gravity, of an object is the point at which the object behaves as if its entire mass is concentrated at that point. If the object is one- or twodimensional, the center of mass is the point at which the object can be balanced horizontally (like a see-saw with riders at either end, in the one-dimensional case). 4

5 For a lamina with its shape defined by a bounded region R 2, and with density given byρ(, y), its center of mass (, y) is located at M y M y where M and M y are the moments of the lamina about the -ais and y-ais, respectively. These are given by M yρ(, y) da, M y ρ(, y) da. These integrals are obtained from the formula for the moment of a point mass about an ais, which is given by the product of the mass and the distance from the ais. Similarly, the moments about the y-, yz- and z-planes, M y, M yz, and M z, of a solid R 3 with density ρ(, y, z) are given by M y zρ(, y, z) dv, M yz ρ(, y, z) dv, M z yρ(, y, z) dv. It follows that its center of mass (, y, z) is located at M yz y M z z M y m. As in the 2- case, each moment is defined using the distance of each point of from the coordinate plane about which the moment is being computed. The moment of interia, or second moment, of an object about an ais gives an indication of the object s tendency to rotate about that ais. For a lamina defined by a region R 2 with density function ρ(, y), its moments of inertia about the -ais and y-ais, I and I y respectively, are given by I y 2 ρ(, y) da, I y 2 ρ(, y) da. On the other hand, for a solid defined by a region R 3 with density ρ(, y, z), its moments of inertia about the coordinate aes are defined by I (y 2 + z 2 )ρ(, y, z) dv, I y ( 2 + z 2 )ρ(, y, z) dv, I z ( 2 + y 2 )ρ(, y, z) dv. The moment I z is also called the polar moment of interia, or the moment of interia about the origin, when reduces to a lamina with density ρ(, y). 5

6 Practice Problems Practice problems from the recommended tetbooks are: Stewart: Section 12.5, ercises 1-19 odd Marsden/Tromba: Section 5.5, ercises 1, 3, 9-25 odd 6