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 21 112 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