Double and Triple Integrals Using Iterated Integrals to find area Using Double Integrals to find Volume Using Triple Integrals to find Volume.

Double and Triple Integrals

•Using Iterated Integrals to find area•Using Double Integrals to find Volume•Using Triple Integrals to find Volume

Three Dimensional Space

In Two-Dimensional Space, you have a circleIn Three-Dimensional space, you have a _____________!!!!!!!!!!!

More 3-D graphs

The Iterated Integral2

1

2y

xydx2

2 2

1 1

2 2x

x y ydydx

Setting up the Double Integral

Finding Area using Double Integrals

Compute the integral on the pictured region

2

R

xy dA

Compute the integral on the pictured region

2

R

xy dA

Finding Volume using the Double Integral

Evaluate the volume using the region

2 21 11

2 2R

x y dA 0 1x 0 1y

Volume using the Triple Integral4 4 4

0 0 0

dV

The cubes density is proportional to its distance away from theXy-plane. Find its mass.

12

0 0 0

x yx

dV