Triple
Integral (Type 1 Solid Region) (example continued)
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Option 1: By inspection of the
intersections, the limits of integration in the x-direction (figure on the left) are x
= 0 to x = y2 . Next “sweep”
the element of area in the y-direction.
In this case the limits of
integration are from y = 0 to y
= 1. So the integral becomes y=1 x= y2 z = 1 – y I =
∫ ∫ ∫ [ f(x,y,z) dz ] dx dy (result) y=0 x=0 z = 0
Option 2: By inspection of the intersections, the limits
of integration in the y-direction (figure on the right) give
y = √x to y = 1 . Then “sweep”
the element of area in the x-direction.
In this case the limits of
integration are from x = 0 to x = 1.
So the integral becomes x=1 y = 1 z = 1 – y I =
∫ ∫
∫
[ f(x,y,z) dz
] dy dx (result) x=0 y=√x z = 0
Click here for a Type 2
Solid Region example. Click here for a Type 3 Solid Region
example. |
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