Multiple IntegralsEx. 2 2

1

2 2x

x y y dy

Ex. 1 2

2 2

1 2

x y dydx

The bounds of the integral represent a region in the xy-plane, and the function represents a height

So the integral represents the volume under the function f (x,y) that lies above the region R in the xy-plane

,R

f x y dA

,d b

c a

f x y dydx Iterated integral

General double integral

Ex.3 2

2

0 1

x y dydx

Ex.2 3

2

1 0

x y dxdy

When R is rectangular, order of integration doesn’t matter.

Ex. Evaluate , where 23R

x y dA , 0 2,1 2R x y x y

Ex. Evaluate , where sinR

y xy dA 1,2 0,R

Ex. Find the volume of the solid bounded by the elliptic paraboloid x2 + 2y2 + z = 16, the planes x = 2 and y = 2, and the coordinate planes.

Ex. Find the average value of f (x,y) = sin x cos y over the region R = [0, ] [0, ].2

2

To evaluate a double integral over a non-rectangular region, we need to describe the region as boundaries of x and y.

Ex. Describe the region R bounded by y = 2x2 and y = 1 + x2.

Ex. Evaluate , where R is the region from

the previous example.

2R

x y dA

Ex. Find the volume that lies below z = x2 + y2 and above the region D in the xy-plane bounded by y = 2x and y = x2.

Ex. Evaluate , where R is the region bounded by

y = x – 1 and y2 = 2x + 6.R

xydA

Ex. Find the volume bounded by x + 2y + z = 2, x = 2y, x = 0, and z = 0

Ex. Evaluate by first switching the order of

integration.

21 21

0 2

x

y

e dxdy

Double Integrals in Polar

When evaluating , it may be

easier to describe R using polar coordinates.

,R

f x y dA

Ex. Describe the region using polar coordinates.

4

2

5

Ex. Describe the region using polar coordinates.

2

-2

• In rectangular coordinates:

dA = dydx

• In polar coordinates:

dA = r drdθ

Ex. Evaluate where R is the region in the

first two quadrants bounded by x2 + y2 = 1 and x2 + y2 = 4.

23 4R

x y dA

Ex. Evaluate

2 2

0 0

a a x

xdydx

Ex. Find the volume of the solid bounded by z = 0 and the paraboloid z = 1 – x2 – y2.

Ex. Find the volume that lies under , above the xy-plane, and inside the cylinder x2 + y2 = 2x.

2 24z x y