ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-01

Transcription

1 ENGI 4430 Multiple Integration Cartesian Double Integrals Page Multiple Integration This chapter provides only a very brief introduction to the major topic of multiple integration. Uses of multiple integration include the evaluation of areas, volumes, masses, total charge on a surface and the location of a centre-of-mass. In future chapters we shall address the issue of integration over non-flat surfaces. Double Integrals (Cartesian Coordinates) Eample 3.01 Find the area shown (assuming SI units).

2 ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-0 Eample 3.01 (continued) Suppose that the surface density on the rectangle is y. Find the mass of the rectangle. The element of mass is OR We can choose to sum horizontally first: m 7 5 y d dy m y ddy 1 The inner integral has no dependency at all on y, in its limits or in its integrand. It can therefore be etracted as a constant factor from inside the outer integral. 5 7 m d y dy 1 which is eactly the same form as before, leading to the same value of 930 kg.

3 ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-03 A double integral f, y da may be separated into a pair of single integrals if D the region D is a rectangle, with sides parallel to the coordinate aes; and f, y g h y. the integrand is separable: D, f y da g h y dy d y 1 y1 y g d h ydy y 1 1 This was the case in Eample Eample 3.0 The triangular region shown here has surface density y. Find the mass of the triangular plate.

4 ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-04 Eample 3.0 (continued) OR We can choose to sum horizontally first (re-iterate): Generally: In Cartesian coordinates on the y-plane, the rectangular element of area is A y. Summing all such elements of area along a vertical strip, the area of the elementary strip is h y y g Summing all the strips across the region R, the total area of the region is: b h A y a y g In the limit as the elements and y shrink to zero, this sum becomes b y h A 1 dy d a y g If the surface density within the region is a function of location, f, y mass of the region is b y h m f, y dy d a y g The inner integral must be evaluated first., then the

5 ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-05 Re-iteration: One may reverse the order of integration by summing the elements of area A horizontally first, then adding the strips across the region from bottom to top. This generates the double integral for the total area of the region y d q y A 1 d dy y c p y The mass becomes y d q y m f, y d dy y c p y Choose the orientation of elementary strips that generates the simpler double integration. For eample, is preferable to

6 ENGI 4430 Multiple Integration Cartesian Double Integrals Page 3-06 Eample 3.03 Evaluate 6 I y da R where R is the region enclosed by the parabola y and the line y. The upper boundary changes form at = 1. The left boundary is the same throughout R. The right boundary is the same throughout R. Therefore choose horizontal strips. 1 y I 6 y d dy y

7 ENGI 4430 Multiple Integration Polar Double Integrals Page 3-07 Polar Double Integrals The Jacobian of the transformation from Cartesian to plane polar coordinates is The element of area is therefore y, r yr r, y da d dy r dr d r Eample 3.04 Find the area enclosed by one loop of the curve r cos. Boundaries: Area:

8 ENGI 4430 Multiple Integration Polar Double Integrals Page 3-08 In general, in plane polar coordinates, D h f, yda f r cos, r sin r dr d g Eample 3.05 Find the centre of mass for a plate of surface density the portion of the circle positive constants. k y, whose boundary is y a that is inside the first quadrant. k and a are Use plane polar coordinates. Boundaries: The positive -ais is the line 0. The positive y-ais is the line /. The circle is r a, which is r = a. Mass: Surface density k y.

9 ENGI 4430 Multiple Integration Polar Double Integrals Page 3-09 Eample 3.05 (continued) First Moments about the -ais:

10 ENGI 4430 Multiple Integration Second Moments Page 3-10 Second Moments Just as the element of first moment of mass about the -ais is M y m y A, the element of second moment of mass about the -ais is the element of second moment of mass about the y-ais is I y A and I A. y Summing all such elements over a plane region R in the limit as A 0, for a region R of surface density, the second moments of mass about the coordinate aes are I y da and R I da y We are usually interested in the second moments about the centroid, y (which is also the centre of mass when the density is constant): and y I y y da R R I da Where possible, we choose coordinate aes that pass through the centre of mass. We can also define the polar moment of mass (or moment of inertia) I I I y da r da y R R This is the moment of inertia of the plane region about an ais, at right angles to the plane, passing through the centre of mass of the region. The SI units of moment of inertia are kg m. The kinetic energy of a rigid body rotating at angular speed about its centre of mass is 1 E I. R

11 ENGI 4430 Multiple Integration Second Moments Page 3-11 Eample 3.06 Find the moment of inertia I for a uniform circular disc of constant surface density and radius a about its centre of mass. By symmetry, the centre of mass of a uniform circular disc is at the geometric centre of the disc. Place the origin there. Use plane polar coordinates. I y da R

12 ENGI 4430 Multiple Integration Second Moments Page 3-1 Eample 3.06 (continued)

13 ENGI 4430 Multiple Integration Second Moments Page 3-13 Eample 3.07 Find the second moments of mass for a uniform rectangle of base b and height h about its centroid. Place the origin at the centre of the rectangle. Use Cartesian coordinates. Note that A bh and = constant. I y da R

14 ENGI 4430 Multiple Integration Second Moments Page 3-14 Eample 3.07 (continued)

15 ENGI 4430 Multiple Integration Second Moments Page 3-15 Two other standard second moments of area about the centroid for constant density are: Triangle bh bh A m I bh mh Semi-circle 1 A a m a I 4 a I 8 9 y and I y a 8 4 The second moment of a collection of regions that share the same centroid is just the sum of the separate second moments. When a region has a hole in it, centered on the centroid of the complete region, then the second moment of the region is the difference between the second moment for the complete region (with the hole filled in) and the second moment for the hole. For eample, the second moments of mass for an annulus of inner radius a and outer radius b are b a b a I Iy and I 4 [using the results from eample 3.06] The mass of the annulus is m b a 4 4 and b a b a b a write, so that we can also m b a I

16 ENGI 4430 Multiple Integration Second Moments Page 3-16 Parallel Ais Theorem The second moment of a composite shape can be found by shifting the reference ais of the standard second moment of each section from the centroidal ais of that section to a parallel ais that passes through the centroid of the composite shape. If the ais passes through the centroid then the second moment I about an ais parallel to the ais and a distance b away from it will be related to I : I y m y A I y da R I Two eamples are in Problem Set 3.

17 ENGI 4430 Multiple Integration Triple Integrals Page 3-17 Triple Integrals The concepts for double integrals (surfaces) etend naturally to triple integrals (volumes). The element of volume, in terms of the Cartesian coordinate system (, y, z) and another orthogonal coordinate system (u, v, w), is, y, z dv d dy dz du dv dw u, v, w and V w v w u v, w w v w u v, w 1 1 1, y, z f, y, zdv f u, v, w, y u, v, w, z u, v, w du dv dw u, v, w The most common choices for non-cartesian coordinate systems in 3 are: Cylindrical Polar Coordinates: cos y sin z z for which the differential volume is dv, y, z d d dz,, z d d dz Spherical Polar Coordinates: rsin cos y rsin sin z r cos for which the differential volume is dv, y, z dr d d r sin dr d d r,,

18 ENGI 4430 Multiple Integration Triple Integrals Page 3-18 Eample 3.08: Verify the formula V a for the volume of a sphere of radius a. Start by placing the origin at the centre of the sphere.

19 ENGI 4430 Multiple Integration Triple Integrals Page 3-19 Eample 3.09: The density of an object is equal to the reciprocal of the distance from the origin. Find the mass and the average density inside the sphere r = a. Use spherical polar coordinates. Density: Mass:

20 ENGI 4430 Multiple Integration Triple Integrals Page 3-0 Eample 3.10: Find the moment of inertia I for a region of uniform density bounded by a sphere of radius a around any ais through the centre of the sphere. Let the z ais be aligned with the rotation ais. Any cross section through the region parallel to the equatorial plane is a circular disc, of radius The mass of the disc is m From eample 3.06, the moment of inertia of this disc is END OF CHAPTER 3