Partial Differentiation To differentiate f x y with respect to x treat y as a constant Examples 1 Let f x y 2x2y3 sin x Then f x fx x y 4xy3 cos x fy x y 6x2y2 f y 2 Let f x y z x2 yz 1 Then fx x y z 2x fy x y z z 1 fz x y z yz 2 3 Let f x y z x2 tan yz2 Then fx 2x tan yz2 fy x2z2 sec2 yz2 fz 2x2yz sec2 yz2 15 Multiple Integrals 15 2 Iterated Integrals Interpretation If R is a region in two dimensions and f is an integrable function on R then where fav is the average value of the f over R In particular cid 82 cid 82 f dA fav Area R R R 1 dA Area R If E is the volume in three dimensions above R and underneath the graph of z f x y then f dA Volume E Otherwise said Volume Average height Area of base Theorem Fubini Suppose R a b c d a rectangle and f continuous Then f x y dA f x y dy dx f x y dx dy cid 90 d cid 90 b c a cid 90 b cid 90 d a c Order evaluate inside integral rst f x y dy dx cid 21 2 Example If R 1 2 0 1 and f x y x2y then x3y x2y dx dy f x y dA cid 20 1 cid 90 1 cid 90 2 cid 90 1 0 7 3 dy y dy 3 x 1 7 6 This is easy for separable functions g x h y dy dx g x dx h y dy Recalling the previous example f gh where g x x2 and h y y whence f x y dA R x2y dx dy x2 dx y dy 7 3 1 2 7 6 cid 90 b cid 90 d a c 0 cid 90 1 cid 90 d cid 90 2 c 1 cid 90 b a 0 1 cid 90 1 cid 90 2 0 1 cid 90 1 0 1 cid 90 cid 90 cid 90 cid 90 cid 90 cid 90 R R cid 90 cid 90 cid 90 b cid 90 cid 90 a R cid 90 d c 15 3 Double Integrals over General Regions Type 1 a x b and g1 x y g2 x Integrate with respect to y rst cid 90 b cid 90 g2 x a g1 x f x y dA f x y dy dx cid 90 cid 90 R cid 90 cid 90 cid 90 cid 90 R cid 90 cid 90 Example For R de ned by 0 x 1 and 3x y 8 4x2 we have cid 90 1 cid 90 8 4x2 4x2 x4 x3 cid 12 cid 12 cid 12 1 3x 0 0 2 cid 90 1 0 cid 12 cid 12 cid 12 y 8 4x2 y 3x cid 90 1 x dA R x dy dx xy dx 8x 4x3 3x2 dx 0 Type 2 h1 y x h2 y and c y d Integrate with respect to x rst cid 90 d cid 90 h2 x c h1 x f x y dA f x y dy dx Example For R de ned by 0 y 1 and y x ey we have cid 90 1 cid 90 ey y cid 90 1 0 x2 cid 12 cid 12 cid 12 x ey x y cid 90 1 dy e2y y2 dy 0 1 2 e2y 1 3 y3 cid 12 cid 12 cid 12 1 0 2x dA R 2x dx dy 0 1 2 e2 1 3 1 2 1 2 e2 5 6 Both Type 1 Type 2 Integrate either way R can be described a x b and g1 x y g2 x or c y d and h1 y x h2 y 2 yxy g1 x y g2 x abRyxx h1 y x h2 y cdRyxRabcd Example Triangle T is a region of type 1 0 x 1 0 y 3 3x and type 2 Hence cid 90 cid 90 0 y 3 y 3 0 x 1 cid 90 1 cid 90 3 3x cid 90 1 y cid 90 3 0 0 3 x dA T x dy dx Or 0 0 x dx dy cid 90 1 cid 90 3 0 0 1 2 cid 17 2 cid 16 3x 3x2 dx 1 2 1 y 3 dy 1 2 Other regions Cut region two create several integrals of either type For example the following region may be sub divided into two regions of type 1 For any function f we have f dA f dA cid 90 cid 90 R cid 90 cid 90 R1 cid 90 cid 90 f dA R2 15 4 Double Integrals in Polar Co ordinates cid 40 cid 40 cid 112 x2 y2 x r cos y r sin cid 33 r tan y x Polar co ordinates In nitessimal Area Starting at x y increase polar co ordinates by in nitessimal amounts dr and d In nitessi mal area is swept out a cid 16 r dr 2 r2 cid 17 d dA 2 since dr 2 cid 28 dr for in nitessimals a dA is the area of a segment between two circles r dr d 3 0123y01xTRR1R2yx d rdrrd dA x y Theorem Suppose that R is a polar rectangle de ned by r1 r r2 and 1 2 and that f is a continuous function of R Then Advanced The Theorem may be modi ed for regions of the plane where g r h for some functions of g h of r the polar equivalent of Type 1 cid 90 2 cid 90 h 1 g f dA R f r cos r sin r dr d A similarly approach can be taken for the analogue of a region of Type 2 cid 90 cid 90 R cid 90 2 cid 90 r2 1 r1 cid 90 cid 90 f x y dA f r cos r sin r dr d 4x 3 dA for the annular region R described R Example Find by 1 x2 y2 4 with x y 0 cid 90 In polar co ordinates R is 1 r 2 and 0 4r2 cos 3r d dr 4x 3 dA cid 90 cid 90 2 R 2 Hence cid 90 2 cid 90 2 cid 90 2 1 1 1 0 4r2 sin 3 r 4r2 r dr 3 2 9 4 cid 12 cid 12 cid 12 2 0 28 3 dr cid 90 cid 90 cid 90 cid 90 Example The three leaved rose has equation r cos 3 Find its area Choose R to be half of one leaf 0 r cos 3 and 0 6 r2 cid 12 cid 12 cid 12 cos 3 r 0 1 2 cid 90 6 cid 90 6 0 d cos 6 1 d cid 90 6 cid 90 6 cid 20 1 0 0 1 …