1 / 30

Chapter 5 Multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 )

Mathematical methods in the physical sciences 3rd edition Mary L. Boas. Chapter 5 Multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 ). Lecture 16 Double & Triple integrals. 1. Introduction.

theola
Download Presentation

Chapter 5 Multiple integrals; applications of integration ( 다중적분 ; 적분의 응용 )

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 5 Multiple integrals; applications of integration (다중적분 ; 적분의 응용) Lecture 16 Double & Triple integrals

  2. 1. Introduction - Use for integration : finding areas, volume, mass, moment of inertia, and so on. - Computers and integral tables are very useful in evaluating integrals. 1) To use these tools efficiently, we need to understand the notation and meaning of integrals. 2) A computer gives you an answer for a definite integral.

  3. 2. Double and triple integrals (이중, 삼중 적분) AREA under the curve VOLUME under the surface “double integral”

  4. - Iterated integrals Example 1.

  5. ‘Integration sequence does not matter.’ (a) (b)

  6.  Integrate with respect to y first,  Integrate with respect to x first,

  7.  Integrate in either order,

  8.  In case of Example 2. mass=? (2,1) density f(x,y)=xy (0,0)

  9.  Triple integral f(x,y,z) over a volume V, Example 3. Find V in ex. 1 by using a triple integral,

  10. Example 4. Find mass in ex. 1 if density =x+z,

  11. 0 1 3. Application of integration; single and multiple integrals (적분의 응용 ; 단일적분, 다중적분) Example 1. y=x^2 from x=0 to x=1 (a) area under the curve (b) mass, if density is xy (c) arc length (d) centroid of the area (e) centroid of the arc (f) moments of the inertia (a) area under the curve (b) mass, if density of xy

  12. (c) arc length of the curve ds dy dx cf. centroid : constant (d) centroid of the area (or arc)

  13. In our example, (e) If  is constant,

  14. (f) moments of the inertia In our example, (=xy)

  15. EX. 2 Rotate the area of Ex. 1 (y=x^2) about x-axis (a) volume (b) moment of inertia about x axis (c) area of curved surface (d) centroid of the curved volume (a) volume (i) (ii)

  16. (b) I_x (=const.) (c) area of curved surface (d) centroid of surface

  17. Mathematical methods in the physical sciences 3rd edition Mary L. Boas Chapter 5 Multiple integrals: applications of integration Lecture 17 Change of variables in integrals

  18. 4. Change of variables in integrals: Jacobians (적분의 변수변환 ; Jacobian) In many applied problems, it is more convenient to use other coordinate systems instead of the rectangular coordinates we have been using. - polar coordinate: 1) Area 2) Curve

  19. Example 1 r=a, density  (a) centroid of the semicircular area

  20. (b) moment of inertia about the y-axis

  21. - Cylindrical coordinate - Spherical coordinate

  22. Jacobians (Using the partial differentiation) ** Prove that

  23. Example 2. z r=h Mass: h y Centroid: x Moment of inertia:

  24. Example 3. Moment of inertia of ‘solid sphere’ of radius a

  25. Example 4. I_z of the solid ellipsoid

  26. In a similar way,

  27. 5. Surface integrals (?) (표면적분) ‘projection of the surface to xy plane’

  28. Example 1. Upper surface of the sphere by the cylinder

  29. H. W. (due 5/28) Chapter 5 2-43 3-17, 18, 19, 20 4-4

More Related