Math 2433 Notes Week Triple Integrals. Integration over an arbitrary solid: Applications: 1. Volume of hypersolid = f ( x, y, z ) dxdydz

Transcription

1 Math 2433 Notes Week Triple Integrals Integration over an arbitrary solid: Applications: 1. Volume of hypersolid = f ( x, y, z ) dxdydz S 2. Volume of S = dxdydz S Reduction to a repeated integral 1. Type I: a x b φ1( x) y φ2( x) ψ1( xy, ) z ψ2( xy, ) 2. Type II: c y d φ ( y) x φ ( y) ψ ( xy, ) z ψ ( xy, ) 3. Type III:

2 Examples: 1. Evaluate: 11+ x xy 4 01 x 0 zdzdydx

3 2. Calculate zdxdydz where T is the tetrahedron in the first octant bounded by the plane T x+ y+ z = 1

4 3. Find the volume of the solid bounded above by the plane y+ z = 2, below by the xy-plane, and on the sides by x = 6 and y = x

5 4. Set up the integral to find the volume of the solid bounded above by the hemisphere z 2 2 = 4 x y and below by the cone z = 3x + 3y 2 2

6 15.7 Cylindrical Coordinates ( xyz,, ) ( r, θ, z) where x= rcosθ r = x + y y y = rsinθ tanθ = x z = z Ω f ( x, y, z) dxdydz = F( r, θ, z) rdrdθdz where Fr (, θ, z) = f( rcos θ, rsin θ, z) Γ We use Cylindrical Coordinates when there is an axis of symmetry, the integrand involves integrating over a circle or part of a circle in the xy -plane. x + y, 2 2 Examples: 1. 2π r π r dzdrdθ

7 2. Evaluate the integral using cylindrical coordinates: 0 x 2, 0 y 4 x, 0 z 16 x y dxdydz where T is the solid formed by T

8 3. Find the volume of the solid bounded above by the cone z 2 = x 2 + y 2, below by the xy-plane, and 2 2 on the sides by the cylinder x + y = 2x.

9 4. Set up the integral to find the volume of the solid bounded above by the hemisphere and below by the cone using cylindrical coordinates

10 15.8 Spherical Coordinates ( xyz,, ) ( ρθφ,, ) where ρ = x + y + z = r + z x = ρsinφcosθ y = ρsinφsinθ ( notice : r = ρsin φ) z z = ρcos φ ( so, cosφ = ) ρ Triple Integrals 2 f ( x, y, z) dxdydz = f ( ρsinφcos θ, ρsinφsin θ, ρcos φ) ρ sinφdρdθdφ T S NOTE: dxdydz ρ 2 sinφdρdθdφ Use when there is spherical symmetry and you are integrating over a sphere or part of a sphere Integrand will involve x + y + z Examples: 1. Find the rectangular coordinates of the point with spherical coordinates π 3π ( ρθφ,, ) = 3,, Find the spherical coordinates, ( ρθφ,, ), of the point with cylindrical coordinates 2 π 1 (, r θ, z) =,, 2 4 2

11 3. Evaluate the integral using spherical coordinates: dxdydz where T is the solid formed by 0 x 1, 0 y 1 x, x + y z 2 x y T

12 4. Convert the integral into spherical coordinates: formed by 0 x 4 y, 0 y 2, 0 z 4 x y z x + y + z dxdydz where T is the solid T

13 5. Set up the integral to find the volume of the solid bounded above by the hemisphere and below by the cone using spherical coordinates

14 15.9 The Jacobian; Changing Variables in Multiple Integration So far, we have used Polar, Cylindrical and Spherical coordinates to make some of our integration problems easier. In calc I, we used u-sub to make some integration easier. We are now going to do something similar but now we will change the xy-coordinates into uv-coordinates. To do that, we will need to transform our region too. Example: Transform the region Ω: y x + = into Γ using the transformation 1 x= 2 u, y = 3v To change our variables in a double integral using our transformation, we first need to find the Jacobian: The Jacobian of the transformation ( xy, ) ( xuv (, ), yuv (, )) is x y u u Juv (, ) = x y v v Example: Find the Jacobian of the transformation x = 4 uv, y = u + 2v 2 2

15 Ω= Now, if the Jacobian is not 0 then the area of J ( u, v) dudv Where Γ represents Γ And so, f ( x, y) dxdy = f ( x( u, v), y( u, v)) J ( u, v) dudv Ω Γ Example: Evaluate ( x + y)cos( π ( x y)) dxdy where Ω is the parallelogram bounded by x y = 0, x y = 3, x+ y = 0, x+ y = 3 First draw Ω: Ω Now, decide on a transformation: Convert and integrate:

16 Another example: Evaluate:

17