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
has the shape of the triangle whose vertices are 0 ,0,0,3,4,0, and whose density is ρ=1+2 x+ y/2. See Example 6.5.4.
Table 6.6.4(a) summarizes the requisite calculations.
m=∫04∫031−x/4ρ ⅆy ⅆx = 25
Ix=∫04∫031−x/4ρ⋅y2 ⅆy ⅆx = 632
Iy=∫04∫031−x/4ρ⋅x2 ⅆy ⅆx = 4885
Table 6.6.4(a) Moments of inertia and radii of gyration
Maple Solution - Interactive
Obtain the equation of the hypotenuse
Tools≻Load Package: Student Precalculus
Context Panel: Student Precalculus≻
Lines and Segments≻Line≻Equation
0,3,4,0→equation of liney=3−34⁢x
Define the density ρ
Context Panel: Assign Name
ρ=1+2 x+ y/2→assign
Obtain the total mass m
Calculus palette: Iterated double-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻m
∫04∫031−x/4ρ ⅆy ⅆx = 25→assign to a namem
Obtain the second moments
Context Panel: Assign to a Name≻Ix (or Iy, as appropriate)
∫04∫031−x/4ρ⋅y2 ⅆy ⅆx = 632→assign to a nameIx
∫04∫031−x/4ρ⋅x2 ⅆy ⅆx = 4885→assign to a nameIy
Obtain the radii of gyration
Expression palette: Square-root template
Press the Enter key.
Context Panel: Approximate≻10 (digits)
→at 10 digits
Maple Solution - Coded
Define the density ρ.
Use the Line command from the Student Precalculus package.
Use the top-level int command.
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