Triple Integrals in Cartesian Coordinates. Triple Integrals in Cylindrical Coordinates. Triple Integrals in Spherical Coordinates

Transcription

1 Chapter 3 Multiple Integral 3. Double Integrals 3. Iterated Integrals 3.3 Double Integrals in Polar Coordinates 3.4 Triple Integrals Triple Integrals in Cartesian Coordinates Triple Integrals in Clindrical Coordinates Triple Integrals in Spherical Coordinates 3.5 Moments and Centre of Mass

2 3. Double Integrals Definition 3. If f is a function of two variables that is defined on a region in the -plane, then the double integral of f over is given b n m f (, ) da lim f (, ) A mn, i j provided this limit eists, in which case f is said to be integrable over. i j

3 Note The double integral of the surface z f (, ) is the volume between the region and below the surface. The sum: i n m j f (, ) is called the double iemann sum and is used as an approimation to the value of the double integral. The double integral inherits most of the properties of the single integral. 3.. Properties of Double Integrals. constant multiple rule i j A cf (, ) da c f (, ) da, c a constant

4 . linear rule [ f (, ) g(, )] da f (, ) da g(, ) da 3. subdivision rule f (, ) da f (, ) da f (, ) da 4. dominance rule, if f (, ) g(, ) f (, ) da g(, ) da

5 3. Iterated Integrals 3.. Evaluating Double Integrals It is impractical to obtain the value of double integral from the definition. We evaluate the integrals b calculating two successive single integrals. d We use the notation f (, ) d to mean that is c held fied and f (, ) is integrated with respect to from c to d. This is called partial integration with respect to. d A( ) f (, ) d c Now we integrate the function A with respect to from, we get: a to b b b d A( ) d f (, ) d d a a c

6 This successive integration process is called iterated integration. f (, ) dd f (, ) d d f (, ) dd f (, ) d d These iterated integrals mean that we first integrate with respect to one variable (while holding the other fied) and then integrating with respect to the other variable while holding the first one fied. It is traditional to omit the brackets and write the iterated integral simpl as f (, ) dd The following theorem gives a practical method for evaluating a double integral b epressing it as an iterated integral.

7 Question In questions a) - c), evaluate the iterated integrals. (a) (b) (c) a r 3 d d sin cos dr d d d

8 Theorem : Fubini s Theorem If f (, ) is continuous over the rectangle : a b, c c, then d b f (, ) da f (, ) dd c a b d a c f (, ) dd Eample Evaluate the integrals. (a) 3 ( 8 ) dd (b) 3 ( 8 ) dd Compare (a) and (b). What can ou sa about the integration? Solution (a) 3 ( 8 ) dd 3 ( 8 ) d d

9 = = d d = (b) 3 ( 8 ) dd = = 3 ( 8 ) d d 3 4 d 3 36d =

10 3.. Nonrectangular egions We limit our stud of double integrals to two basic tpes of regions: Tpe I and Tpe II. Definition (a) A plane region is said to be of Tpe I if it lies between the graphs of two continuous functions of. (, ): a b, g( ) g( ) (b) A plane region is said to be of Tpe II if it lies between the graphs of two continuous functions of. (, ): h( ) h( ), c d

11 Tpe I egion - integrating first with respect to Tpe I (Vertical Strip): fied between a and b, varies from g ( ) to g ( ). Tpe II egion - integrating first with respect to Tpe II (Horizontal Strip): fied between c and d, varies from h ( ) to h ( ).

12 Theorem (a) If is a Tpe I region, then ( ) f (, ) da f (, ) dd a g ( ) (b) If is a Tpe II region, then b g ( ) f (, ) da f (, ) dd c h ( ) d h Eample Evaluate ( ) da over the region enclosed b the lines. Solution, and Sketch the region: set up the limits of integration

13 Choose order of integration: Tpe I, fied ) ( ) ( dd da = d d 4 = Alternativel, reversing the order of integration: Tpe II, fied = = =

14 ) ( ) ( dd da = d d 8 5 = = = /

15 3..3 Double Integral as Area and Volume Definition (a) The area of the region in the -plane is given b A da (b) If f is continuous and f (, ) on the region, the volume of the solid under the surface z f (, ) above the region is given b Eample V f (, ) da Find the area of the region bounded b and in the first quadrant.

16 Solution Sketch the region: = = Order of integration: Tpe I, fied Area = dd d = 3 d 3 unit 6

17 Question In questions (a) - (c), evaluate f, da. f, (a) ;, :,. (b) f, 4 where is the closed rectangular region with vertices (,), (3,), (,4) and (-,4). f, 3 (c) 3 ;, :,. Question In questions (a) - (b), sketch the closed region bounded b the given curves, and find the area of the region using a double integral. (a),,, 4. (b),,, 3.

18 Question 3 In questions 3(a) - 3(b), sketch the solid in the first octant bounded b the given surfaces, and find its volume b using a double integral. (a) z 4,,, z. (b) z 4,,,, z.

19 3.3 Double Integral in Polar Form 3.3. Polar Coordinates Sstem A polar coordinate sstem consist of a fied point O called the origin or pole and a line segment starting from the pole called the polar ais. adial ais r P(r, ) Polar ais r radial coordinate polar angle Definition Polar coordinates of a point P is written as r, where r is the distance of P from the pole and is the angle measured from the polar ais to the line OP (radial ais).

20 3.3. elationship between Polar and Cartesian Coordinates P(r, ) r r cos r sin, r O tan Note (i) Polar coordinate of a point is not unique. (ii) is positive in an anticlockwise direction, and negative if it is taken clockwise. (iii) A point r, is in the opposite direction of point r,.

21 3.3.3 Integrals in Polar Coordinates If is a circular region (involves easil described using polar coordinates. ), it is Divide the region into polar rectangles. r = r = O Find the area of tpical polar rectangle: r A k O = area of large sector area of small sector k r k r r k r r r k k k

22 Alternativel: If the mesh is small enough, we can assume that, r r r and with this assumption we can also assume that our polar slab is close enough to a rectangle, A r r Thinking of volume, we make the equation z f ( r cos, r sin ), thus the iemann sum can be written as:

23 V m n i j f * i * j * ( r, ) r r Taking the limit we have the actual volume, f (, ) da f ( r, ) r drd A version of Fubini s Theorem now sas that the integral can be evaluated b iteration with respect to r and. Theorem Let be a simple polar region whose boundaries are the ras and the curves r ( ) r f ( r, ) is continuous on, then and r and r ( ) ( ) f (, ) da f ( r, ) r drd r r r r ( ). If

24 3.3.4 Finding limits of Integration Eample Find the limits of integration for integrating f ( r, ) over the region that lies inside the cardiod r cos and outside the circle r =. Solution Step : Sketch

25 = / r = r = + cos = / Step : the r-limits of integration A tpical ra from the origin enters where r = and leaves where r cos. Step 3: the -limits of integration The ras from the origin that intersect run from to The integral is.

26 cos f ( r, ) r drd Note We ma, of course, integrate first with respect to and then with respect to r if this is more convenient Changing Cartesian Integrals into Polar Integrals The procedure for changing Cartesian integral f (, ) da into a polar integral has two steps. r cos, r sin and replace dd b r drd in the Step : Substitute Cartesian integral.

27 Step : Suppl polar limits of integration for the boundar of. The Cartesian integral then becomes r ( ) f ( r, ) da f ( r cos, r sin ) rdrd r ( ) Eample Evaluate ( ) da where is the region inside the circle 4. Solution We evaluate the integral in polar form. KNOW: egion : 4 r r 4 or r =

28 r = = = ( ) da ( r ) r drd

29 Question In questions (a) and (b), evaluate the double integral. (a) (b) sin r a sin Question cos r drd drd Sketch the closed region bounded b the polar equations, and find its area b using a double integral in polar coordinates. (a) r, r 4 sin,,. 3 3 (b) The region inside the cardiod r sin and outside the circle r 3.

30 Question 3 Evaluate the integrals b changing to polar coordinates. e dd Question 4 Find the volume of the solid bounded b z 9 and z 5.