MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS

Transcription

1 MATH WEEK 9 - WEDNESDAY TRIPLE INTEGRALS

2 MATH GOALS Be able to set up and evaluate triple integrals using rectangular, cylindrical, and spherical coordinates

3 MATH TRIPLE INTEGRALS We integrate functions of three variables over three dimensional solids S F (x, y, z) dv Chop the solid S up into a bunch of cubes with volume dv Pick a point in each cube and evaluate F there Add up all of these products (F dv) S (x,y,z ) dv

4 MATH INTERPRETATIONS If we think of F(x,y,z) as giving the density of the solid S at (x,y,z), then the triple integral gives us the mass of S If F(x,y,z), then the integral gives us the volume of S S (x,y,z ) dv

5 MATH EXAMPLE /3 d dy cos(xy) sin(xy) dy zxsin(xy) dzdydx x sin(xy) x cos(xy)+c /3 /3 /3 /3 /3 z x sin(xy) z z x sin(xy) dydx y x x cos(xy) y cos(xy) y dx cos( x)+ dx sin( x)+ x / y dydx dx

6 MATH LOTS OF WAYS TO SETUP Let s set up a few triple integrals for the volume of the solid bounded by y + z and y x in the first octant This means, we ll just integrate F(x,y,z) Here s what the solid looks like: A sketch will really help with these problems

7 MATH Let s say we want to integrate in the order dzdydx Once we integrate with respect to z, z is gone Visually, we can think of flattening the solid onto the xy-plane The top bound for z is the surface z (-y ) / The bottom bound is the xy-plane, z

8 MATH Once we ve flattened out in the z-direction, we have a double integral to set up, which we already know how to do! We have y x & y and x & x So the triple integral becomes x y dzdydx

9 MATH Alternatively, we could have gone with dzdxdy In this case all that changes is the outer double integral Going back to the flattened image on the xy-plane, we get x & x y and y & y y y dzdxdy

10 MATH We could also not start with z. For example, let s try dxdzdy Integrating with respect to x first will flatten the picture onto the yz-plane On top (meaning further out towards us), we have x y On the bottom (meaning further back) we have x yz-plane

11 MATH Now we just set up the bounds for the outer two integrals based on the flattened image on the yzplane z & z (-y ) / y & y So the triple integral becomes y y dxdzdy

12 MATH Pick one of these three to integrate: x y dzdydx y y y dzdxdy With the dzdydx integral, we end up needing trig substitution to perform the second integration, so we should go with the second or third option y dxdzdy

13 MATH y y dzdxdy y y z y x y y dxdy y dxdy y y dy udu 3 u3/ dy 3

14 MATH EXAMPLE Evaluate z ydv where G is the solid enclosed by z y, y x, y 4, and z. G

15 MATH Let s try dzdydx first z and z y Flatten the solid onto the xyplane Now for y we have y x & y 4 Finally, x - & x

16 MATH 4 x WE HAVE TO BE CAREFUL WHEN USING SYMMETRY: y z ydzdydx IT WORKS HERE BECAUSE BOTH THE FUNCTION WE RE INTEGRATING AND THE REGION OVER WHICH WE RE INTEGRATING ARE SYMMETRIC OVER THE PLANE X x 4 x 4 y x 4 y z z y ydzdydx y ydydx x y 5/ dydx 7 y7/ 4 x dx 8 x 7 4 8x 8 x8 x dydx dx 7 (56 3) 64

17 MATH Let s look at some alternative setups We could have started with x instead and done dxdzdy If we draw a line through the solid in the x-direction, it first hits the back half of the parabolic surface and then the front half of the parabolic surface If we then collapse the picture onto the yz-plane, we get this 4 y y y z ydxdzdy

18 MATH TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES Cylindrical coordinates We already know from polar that da rdrdθ So, for dv we get rdrdθdz Just replace dxdy or dydx with rdrdθ Spherical coordinates For now, let s just accept that in spherical coordinates, dv becomes ρ sinφdρdφdθ We ll come back to why this is the case in the next section

19 MATH EXAMPLE 4 x 4 x y Consider the integral x + y dzdydx 4 x y First let s get a sense of what the region/solid looks like z -(4 - x - y ) / and z (4 - x - y ) / Squaring both sides of either equation, we get a sphere of radius centered at (,,): x + y + z 4 So we re going from the bottom half of the sphere to the top half y & y (4-x ) / and x & x On the xy-plane, we go from the line y to the top half of a circle of radius, but only from x to x.

20 MATH

21 MATH Setup in cylindrical coordinates Since z is common to rectangular and cylindrical, let s start with that z 4 x y z 4 r z 4 x y z 4 r Now we can look at what remains on the xy-plane and convert that to polar (recall: cylindrical polar + z) y & y (4-x ) / and x & x On the xy-plane, we go from the line y to the top half of a circle of radius, but only from x to x. r goes from to and θ goes from to π/

22 MATH z 4 r θ π r z 4 r r θ 4 x 4 x y 4 x y (x + y ) dzdydx π/ 4 r 4 r r rdzdrdθ

23 MATH For spherical, let s start with ρ: The sphere of radius is simply ρ The region starts at the origin: ρ Remember, φ measures the angle taken from the positive z-axis In order to cover the quarter-sphere, φ needs to go from to π. We already know what θ does from cylindrical coordinates The integrand (the function we re integrating) is a little more involved

24 MATH x + y ( sin cos ) +( sin sin ) sin cos + sin sin sin (cos +sin ) sin Lastly, we can t forget about the extra term, ρ sinφ: 4 x 4 x y 4 x y (x + y ) dzdydx / sin sin d d d / 4 sin 3 d d d