Chapter 6: Applications of Double Integration
Section 6.6: Second Moments
Find the moments of inertial Ix and Iy, the total mass m, and the radii of gyration Rx and Ry of the lamina that occupies R, the region bounded by y=1−x2 and the x-axis. Take ρ=1. See Example 6.5.1.
Table 6.6.1(a) summarizes the requisite calculations.
m=∫−11∫01−x21 ⅆy ⅆx = π/2
Ix=∫−11∫01−x2y2 ⅆy ⅆx = π/8
Iy=∫−11∫01−x2x2 ⅆy ⅆx = π/8
Rx=Iy/m=π/8π/2 = 12
Ry=Ix/m=π/8π/2 = 12
Table 6.6.1(a) Moments of inertia and radii of gyration
Maple Solution - Interactive
Obtain the total mass m
Iterated double-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻m
∫−11∫01−x21 ⅆy ⅆx = 12⁢π→assign to a namem
Obtain the second moments
Calculus palette: Iterated double-integral template
Context Panel: Assign to a Name≻Ix (or Iy, as appropriate)
∫−11∫01−x2y2 ⅆy ⅆx = 18⁢π→assign to a nameIx
∫−11∫01−x2x2 ⅆy ⅆx = 18⁢π→assign to a nameIy
Obtain the radii of gyration
Expression palette: Square-root template
Press the Enter key.
Maple Solution - Coded
Use the top-level int command.
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