1. Chapter 5 Multiple integrals; applications of integration
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.’
(b)

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

7.  Integrate in either order,

8.  In case of
Example 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. 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 1
(a) area under the curve
(b) mass, if density of xy

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

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. Chapter 5 Multiple integrals: applications of integration
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’

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

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