double and triple integrals
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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
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