∫∫ ∫ ∫ ∫∫ - UH

Math 2433 - Week 6 Notes (part 2)

16.4 The Double Integral as a Limit of Riemann Sums; Polar Coordinates

Given F = F (r, ) continuous on

: a r b,

b

F(r, )r drd = F(r, )r drd

a

And the double integral of F (r, ) over the polar region

: , 1 ( ) r 2 ( )

2 ( )

F(r, )r drd = F(r, )r drd

1( )

Rectangular

A = dxdy V = f (x, y)dxdy

Polar

b

A = r drd a

b

V = F(r, )r drd a

5. Find the volume of the solid bounded above by the paraboloid z = x2 + y2 , below by the xy-plane and on the sides by the cylinder x2 + y2 = 2y

Popper 06

6. Which of the following will represent the volume of the solid bounded above by the plane z = x + 4, below by the xy-plane, and on the sides by the circular cylinder x2 + y2 = 9.

2 3

a) (r sin + 4) rdrd 00

2 3

b) (r cos ) rdrd 00

r cos +4

c) rdrd 00

2 3

d) (r cos + 4)rdrd 00

16.5 Applications of Double Integration

Volume Area Mass

Center of Mass

Rectangular

f (x, y)dx dy

1dx dy

M = (x, y)dx dy, = density function

(x, y) x dx dy

(x, y) y dx dy

xM =

M

, yM =

M

Polar

F(r, )r dr d

r dr d

~

Example: Find the mass and center of mass where is the triangle with vertices (0,0), (1, 3) and (1, 5). (x, y) = xy

16.6 Triple Integrals

The biggest difference between f (x, y)dxdy and f (x, y, z)dxdydz is that instead of working with

T

two variables continuous over a plane, we are working with three variables over a continuous three

dimensional space.

Integration over a "Box":

Given f = f (x, y, z) is continuous on the rectangular "box" B, where B : a1 x b1, a2 y b2, a3 z b3

16.7 Triple Integrals Integration over an arbitrary solid:

Applications:

1. " Volume" of "hypersolid" = f (x, y, z)dxdydz S

2. Volume of S = S dxdydz

Reduction to a repeated integral

1. Type I: a x b 1 (x) y 2 (x) 1 (x, y) z 2 (x, y) 2. Type II: c y d 1 ( y) x 2 ( y) 1 (x, y) z 2 (x, y)

3. Type III: 4. ......

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