Now we are looking to find a volume of solid S that lies below a surface z = f(x,y) and R= ab, cd,,[a,b] is the interval over

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Multiple Itegratio Double Itegrals, Volume, ad Iterated Itegrals I sigle variable calculus we looked to fid the area uder a curve f(x) bouded by the x- axis over some iterval usig summatios the that

1 Multiple Itegratio Double Itegrals, Volume, ad Iterated Itegrals I sigle variable calculus we looked to fid the area uder a curve f(x) bouded by the x- axis over some iterval usig summatios the that led to usig itegrals. That same process ca be traslated over to multi-variable calculus ad volume. Istead of calculatig the areas of a rectagles we will be calculatig the volume of rectagular prisms. b b a Sigle Variable : lim f ( ci ) x= f ( x) dx a = where x= i Now we are lookig to fid a volume of solid S that lies below a surface z = f(x,y) ad = ab, cd,,[a,b] is the iterval over above some rectagle i the x y plae where [ ] [ ] the x axis ad [c,d] is the iterval over the y axis. We begi by sub-divig ito sub-rectagles. So [a,b] is divided ito sub-itervals b a x= ad [c,d] as well d c y=. Every sub-rectagle should have area: A= x y The the height of each rectagular prism is z =f(x i,y i ). So to fid the volume of each rectagular prism we use V = f ( x, y ) A i i Now if we wat to add all these prisms up we use the sum i= f ( x, y ) A i i If we thik about the iema sum from calculus I, we ca take the limit of this sum as ( = the umber of sub-rectagles) or let the diagoal of each sub-rectagle go to zero) ad this will give the actual volume ad ca be foud usig a itegral (or two).

2 Double Itegrals: If f is defied o a closed, bouded regio i the xy-plae, the the double itegral of f over is give by f ( x, y) da= lim f ( x, y ) A i = i i provided the limit exists. If z =f(x,y) 0 for all (x,y) i, the the above double itegral will represet the volume of the solid that lies above ad below f. All the usual properties of sigle itegrals apply to double itegrals I order to evaluate the double itegral we ca express them as iterated itegrals Iterated Itegrals: To apply the Fudametal Theorem of Calculus to double itegrals we use the followig methods called iterated itegrals b d d b b d d b f ( x, y) dy dx= f ( x, y) dx dy= f ( x, y) dydx= f ( x, y) dxdy a c c a a c c a This process ca be thought of as itegratig oe variable at a time. emember ot every iterated itegral has to represet area. Fubii s Theorem: The double itegral of a cotiuous fuctio f(x,y) over a rectagle = [a,b]*[c,d] is equal to the iterated itegral b d f ( x, y) da= f ( x, y) dydx= f ( x, y) dxdy d b a c c a ( x y ) da if = { x, y } (x+ 3 y ) da if = { x 3, y 4} Ex: Fid the volume betwee the graph of f ( x, y) = 6 x 3y ad the rectagle = [0,3] [0,] Whe a fuctio is a product f ( x, y) = g( x) h( y) the double itegral over a rectagle [a,b]*[c,d] is simply the product of the sigle itegrals. b d ( g xdx a )( h y dy c ) f ( x, y) da= g( x) h( y) dxdy= ( ) ( ) y da x+ if = { 0 x,0 y 4}

3 Double Itegrals over More Geeral egios: What if the regio uder the surface is ot a rectagle but istead a more geeral regio? Ca we still fid its volume? -----YES!!! (this is more commo) We redefie the fuctio to be itself i the regio ad zero outside the regio by the followig f ( x, y) if ( x, y) D F( x, y) = 0 if ( x, y) but ( x, y) D where D is the domai of the regio I order for this redefiig to work ad actually fid the volume, oe set of limits of itegratio must be costat. We set it up so that the iside limits ca be fuctios of the opposite variable ad the outside limits are umerical. i. Give f ( x, yda ) ( x, y ) a x b, g ( x ) y h ( x ) such that { } b h( x) a g ( x) = the f ( x, yda ) = f ( x, ydydx ) if the regio is vertically simple such that {( x, y ) g ( y ) x h ( y ), c y d } ii. Give f ( x, yda ) d h( y) c g( y) = the f ( x, yda ) = f ( x, ydxdy ) if the regio is horizotally simple x (see what happes if you switch the order of itegratio) 4 ye x dydx Ex: Fid the volume of the solid regio bouded by the paraboloid z= 4 x y ad the xy-plae.

Triple Itegrals We defie sigle itegrals for fuctios of sigle variables ad double itegrals for fuctios of two variables. Now we defie triple itegrals for fuctios of three variables.

The, ecompass B with a etwork of boxes ad form the ier partitio cosistig of all boxes lyig etirely withi B.

poit i each box i= f ( x, y, z ) V i i i The by lettig the umber of boxes go to ifiity ( or the diagoal of each box go to zero) we ca evaluate the volume.

4 Triple Itegrals We defie sigle itegrals for fuctios of sigle variables ad double itegrals for fuctios of two variables. Now we defie triple itegrals for fuctios of three variables. Cosider the a fuctio w= f ( x, y, z) that is cotiuous o a rectagular box, B {( x, y, z) a x bc, y d, e z f} =. The, ecompass B with a etwork of boxes ad form the ier partitio cosistig of all boxes lyig etirely withi B. The volume of each sub-box is V = x y z If we start stackig the boxes up to the fuctio we ca use the eima sum to approximate the volume by addig up the sub-boxes where (x i, y i, z i ) is a sample poit i each box i= f ( x, y, z ) V i i i The by lettig the umber of boxes go to ifiity ( or the diagoal of each box go to zero) we ca evaluate the volume. Triple Itegrals: If f is cotiuous over a bouded solid regio B, the the triple itegral of f over B is defie by provided the limit exists. B f ( x, y, z) dv = lim f ( x, y, z ) V i = i i i Fubii s Theorem for Triple Itegrals: The triple itegral of a cotiuous fuctio f(x, y, z) over a box B is equal to the iterated itegral B b d f ( x, y, z) dv = f ( x, y, z) dzdydx f a c e this iterated itegral ca be evaluated i ay order. ( xyz ) dv where B= {( x, y, z) 0 x, y,0 z 3} B What do we do whe the regio is more geeralized the a box? (Meaig the limits of itegratio are fuctios istead of umbers)

5 More Geeral egios for Triple Itegrals i 3-D Space: Let f be cotiuous o a solid regio B defied by a x b, h ( x) y h ( x), g( x, y) z g( x, y) (simple with respect to the order dzdydx) Where h, h, g, ad g are cotiuous fuctios. The, B b h ( x) g ( x, y) f ( x, y, z) dv = f ( x, y, z) dzdydx a h ( x) g ( x, y) If this regio is simple i differet orders it would work out i a similar patter accordigly. 9 y y 9x zdzdxdy Ex: Itegrate f ( x, y, z) = z over the regio lyig below the upper hemisphere of radius 3 ad the above the triagle i the xy-plae bouded by the lie x =, y = 0 ad x = y. Ex: Fid the volume of the solid i the octat x, y, z 0 bouded by the plaes x+ y+ z= ad x+ y+ z= Itegratio i Polar, Cylidrical, ad Spherical Coordiates If f(x,y) is a cotiuous fuctio ad you eed to itegrate it over a circle at the origi, the is much easier to use polar coordiates. ecall: To covert from polar to rectagular: x = rcosθ ; y = rsiθ To covert from rectagular to polar: r = x + y ; Θ = arcta(y/x) To defie a double itegral of a cotiuous fuctio z= f ( x, y) i polar coordiates, cosider a regio bouded by the graphs of r= g ( θ ) ad r ( ) = g θ ad the lies θ = α ad θ = β. Istead of partitioig ito small rectagles, use a partitio of small polar sectors. O superimpose a polar grid made of rays ad circular arcs. The area of a specific polar sectors lyig etirely withi is A= r r θ, where r= r r ad θ = θ θ This implies that the volume of a solid with height f ( r cos θ, r si θ ) r r θ ad we have β g ( θ ) f r θ r θ da f r θ r θ rdrd θ ( cos, si ) = ( cos, si ) α g ( θ ) where g( θ ) r g( θ ) ad α θ β

6 Ex: Let be the regio that lies betwee the circles x + y = ad x + y = 5. Evaluate ( x + y) da. Ex: Let be the regio that lies betwee the y 0 ad x + y. Evaluate 3 ( y( x + y ) ) da usig polar. Ex: Fid the volume of the solid that lies above the circle x + y 4i the xy-plae ad below the hemisphere z= 6 x y Cylidrical Coordiates: Cylidrical coordiates are useful whe the domai has axial symmetry, symmetry with respect to a axis. Usually whe we have the expressio x + y. ecall: x = rcosθ y = rsiθ z = z To set up the triple itegral i cylidrical coordiates we assume that the domai of itegratio W ca be described as the regio betwee two surfaces z( r, θ ) θ z( r, θ ) lyig over a domai D i the xy-plae with polar descriptio D : θ θ θ, r ( θ ) θ r ( θ ) A triple itegral over W ca be writte as W θ r ( θ ) z ( r, θ ) f x y z dv f r θ r θ z rdzdrdθ (,, ) = ( cos, si, ) θ r ( θ ) z ( r, θ ) by covertig to cylidrical coordiates: ( x + y ) dv where { (,, ), 4 4, } W = x y z x x y x x + y z. W

7 Spherical Coordiates: ecall: ectagular Spherical : x = ρsiφcosθ, y = ρsiφsiθ, z = ρcosφ Spherical ectagular: ρ = x + y + z, taθ = y/x, cosφ = z/ρ As we saw the chage of variables formula i cylidrical coordiates ca be summarized by dv = rdrdθdz. With spherical the same thigs ca be doe to summarize usig dv = ρ φdρdφdθ si By workig with a the regio kow as a spherical wedge {(,,,, } W = ρ φ θ θ θ θ φ φ φ ρ ρ ρ. The triple itegral over w ca be writte as W θ φ ρ ( θφ, ) f dv f ρ θ φ ρ θ φ ρ φ ρ φdρdφdθ ( ρcosθ si φ, ρsiθsi φ, ρcos θ) = ( cos si, si si, cos ) si θ φ ρ ( θφ, ) ( x + y + z ) 3 e dv where W = sphere with radius. W Chage of Variables: Jacobias Whe we covert from rectagular to aother system a extra factor appears for our value of dv (ie. rect. polar gives a extra factor r) Where do these extra factors come from????????? We ca trasform rectagular coordiates to ay system this is logical by chagig the variable. Whe we do this we must also adjust the differetials da or dv. The adjustmet/extra factors come from the chage i variables ad are called the Jacobia. The Jacobia: If x = g(u, v) ad y = h(u, v), the the Jacobia of x ad y with respect to u ad v, deoted by ( x, y) ( u, v), is x x ( x, y) u v x y x y = = ( u, v) y y u v v u u v I three idepedet variable this exteds to a 3X3 matrix Ex: Fid the Jacobia for x= ( u v), y= ( u+ v) Ex: Fid the Jacobia for x = rcosθ, y = rsiθ

8 Chage of Variables for Double Itegrals Let be a vertically or horizotally simple regio i the xy-plae, ad let S be a vertically or horizotally simple regio i the uv-plae. Let T from S to be give by T(u, v) = (x, y) =(g(u, v),h(u, v)), where g ad h have cotiuous first partial derivatives. Assume the T is oe-to-oe except possibly o the boudary of S. If f is cotiuous o, ad ( x, y) ( u, v) is ozero o S, the ( x, y) f ( x, y) dxdy= f ( g( u, v), hu (, v))) dudv ( u, v) Ex: Use the trasformatio s x u v = + ad y uv v regio bouded by y = x, y = x, y = 3 x ad y = 6 x. = to compute ( x+ y) dxdy + where D is the D

Math 21B-B - Homework Set 2

Math 21B-B - Homework Set 2 Math B-B - Homework Set Sectio 5.:. a) lim P k= c k c k ) x k, where P is a partitio of [, 5. x x ) dx b) lim P k= 4 ck x k, where P is a partitio of [,. 4 x dx c) lim P k= ta c k ) x k, where P is a partitio