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

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.

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

- Iterated integrals Example 1.

‘Integration sequence does not matter.’ (b)

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

 Integrate in either order,

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

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

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

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

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

In our example, (e) If  is constant,

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

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)

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

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

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

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

(b) moment of inertia about the y-axis

- Cylindrical coordinate - Spherical coordinate

Jacobians (Using the partial differentiation) ** Prove that

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

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

Example 4. I_z of the solid ellipsoid

In a similar way,

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

Example 1. Upper surface of the sphere by the cylinder

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