Chapter 8: Applications of Triple Integration
Section 8.1: Volume
Use an iterated triple integral to obtain the volume of R, the region enclosed by the cylinders z=4−x2 and z=5 x2, and the planes y=0 and x+y=2/3.
Figure 8.1.9(a) shows the solid whose volume is obtained by iterating a triple integral in Cartesian coordinates in the order dy dz dx.
∫−2/32/3∫5 x24−x2∫02/3−x1 dy dz dx = 329
Figure 8.1.9(a) The region R
Figure 8.1.9(b) The face y=0
The integration is from the blue face (y=0) to the red face (y=2/3−z); then from the green surface (z=5 x2) to the gold surface (z=4−x2); and finally, from x=−2/3 to x=2/3.
The bounds x=±2/3 are found by intersecting the upper and lower surfaces z=4−x2 and z=5 x2, respectively. (See Figure 8.1.9(b).) This leads to the equation 4−x2=5 x2, and to the solution x=±2/3.
Maple Solution - Interactive
Tools≻Load Package: Student Multivariate Calculus
Access the MultiInt command via the Context Panel
Type the integrand, 1.
Context Panel: Student Multivariate Calculus≻Integrate≻Iterated
Fill in the fields of the two dialogs shown below.
Context Panel: Evaluate Integral
Table 8.1.9(a) provides a solution by a task template that integrates in Cartesian coordinates and draws the region of integration.
Calculus - Multivariate≻Integration≻Visualizing Regions of Integration≻Cartesian 3-D
Evaluate ∭RΨx,y,z dv and Graph R
Volume Element dv
Select dvdz dy dxdz dx dydx dy dzdx dz dydy dx dzdy dz dx
, where Ψ=
Table 8.1.9(a) Task template integrating in Cartesian coordinates
Table 8.1.9(b) provides a solution from first principles.
Iterated triple-integral template
Context Panel: Evaluate and Display Inline
∫−2/32/3∫5 x24−x2∫02/3−x1 ⅆy ⅆz ⅆx = 329
Table 8.1.9(b) Integration via first principles
Maple Solution - Coded
Table 8.1.9(c) obtains a solution via the MultiInt command in the Student MultivariateCalculus package.
Install the Student MultivariateCalculus package.
MultiInt1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3 = 329
Table 8.1.9(c) MultiInt command iterating in Cartesian coordinates in the order dy dz dx
Table 8.1.9(d) implements the iterated integration via the top-level Int and int commands.
Int1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3=int1,y=0..2/3−x,z=5 x2..4−x2,x=−2/3..2/3
Table 8.1.9(d) Top-level Int and int commands
Notice how Maple rewrites the radicals in the limits of integration.
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