7.6E
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- Nov 19, 2020
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( \newcommand{\kernel}{\mathrm{null}\,}\)
Exercise 1: Mass
In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function ρ.
1. R is the triangular region with vertices (0,0), (0,3), and (6,0); ρ(x,y)=xy.
- Answer
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272
3. R is the triangular region with vertices (0,0), (1,1), and (0,5); ρ(x,y)=x+y.
4. R is the rectangular region with vertices (0,0), (0,3), (6,3) and (6,0); ρ(x,y)=√xy.
- Answer
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24√2
5. R is the rectangular region with vertices (0,1), (0,3), (3,3) and (3,1); ρ(x,y)=x2y.
6. R is the trapezoidal region determined by the lines y=−14x+52, y=0, y=2, and x=0; ρ(x,y)=3xy.
- Answer
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76
7. R is the trapezoidal region determined by the lines y=0, y=1, y=x and y=−x+3; ρ(x,y)=2x+y.
8. R is the disk of radius 2 centered at (1,2); ρ(x,y)=x2+y2−2x−4y+5.
- Answer
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8π
10. R is the unit disk; ρ(x,y)=3x4+6x2y2+3y4.
11. R is the region enclosed by the ellipse x2+4y2=1; ρ(x,y)=1.
- Answer
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π2
12. R={(x,y)|9x2+y2≤1, x≥0, y≥0}; ρ(x,y)=√9x2+y2.
13. R is the region bounded by y=x, y=−x, y=x+2, y=−x+2; ρ(x,y)=1.
- Answer
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2
14. R is the region bounded by y=1x, y=2x, y=1, and y=2; ρ(x,y)=4(x+y).
Exercise 2 (CAS)
In the following exercises, consider a lamina occupying the region R and having the density function ρ given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions.
a. Find the moments Mx and My about the x-axis and y-axis, respectively.
b. Calculate and plot the center of mass of the lamina.
c. [T] Use a CAS to locate the center of mass on the graph of R.
1. [T] R is the triangular region with vertices (0,0), (0,3), and (6,0); ρ(x,y)=xy.
- Answer
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a. Mx=815, My=1625; b. ˉx=125, ˉy=65;
c.
2. [T] R is the triangular region with vertices (0,0), (1,1), and (0,5); ρ(x,y)=x+y.
3. [T] R is the rectangular region with vertices (0,0), (0,3), (6,3), and (6,0); ρ(x,y)=√xy.
- Answer
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a. Mx=216√25, My=432√25; b. ˉx=185, ˉy=95;
c.
4. [T] R is the rectangular region with vertices (0,1), (0,3), (3,3), and (3,1); ρ(x,y)=x2y.
[5. T] R is the trapezoidal region determined by the lines y=−14x+52, y=0, y=2, and x = 0; \space \rho (x,y) = 3xy\).
- Answer
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a. Mx=3685, My=15525; b. ˉx=9295, ˉy=38895;
c.
6. [T] R is the trapezoidal region determined by the lines y=0, y=1, y=x, and y = -x + 3; \space \rho (x,y) = 2x + y\).
7. [T] R is the disk of radius 2 centered at (1,2); ρ(x,y)=x2+y2−2x−4y+5.
- Answer
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a. Mx=16π, My=8π; b. ˉx=1, ˉy=2;
c.
7. [T] R is the unit disk; ρ(x,y)=3x4+6x2y2+3y4.
8. [T] R is the region enclosed by the ellipse x2+4y2=1; ρ(x,y)=1.
- Answer
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a. Mx=0, My=0); b. ˉx=0, ˉy=0;
c.
9. [T] R={(x,y)|9x2+y2≤1, x≥0, y≥0}; ρ(x,y)=√9x2+y2.
10. [T] R is the region bounded by y=x, y=−x, y=x+2, and y=−x+2; ρ(x,y)=1.
- Answer
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a. Mx=2, My=0); b. ˉx=0, ˉy=1;
c.
11. [T] R is the region bounded by y=1x, y=2x, y=1, and y=2; ρ(x,y)=4(x+y).
Exercise 3
In the following exercises, consider a lamina occupying the region R and having the density function ρ given in the first two groups of Exercises.
a. Find the moments of inertia Ix, Iy and I0 about the x-axis, y-axis, and origin, respectively.
b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively.
1. R is the triangular region with vertices (0,0), (0,3), and (6,0); ρ(x,y)=xy.
- Answer
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a. Ix=24310, Iy=4865, and I0=2432; b. Rx=3√55, Ry=6√55, and R0=3
2. R is the triangular region with vertices (0,0), (1,1), and (0,5); ρ(x,y)=x+y.
3. R is the rectangular region with vertices (0,0), (0,3), (6,3), and (6,0); ρ(x,y)=√xy.
- Answer
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a. Ix=2592√27, Iy=648√27, and I0=3240√27; b. Rx=6√217, Ry=3√217, and R0=3√1067
4. R is the rectangular region with vertices (0,1), (0,3), (3,3), and (3,1); ρ(x,y)=x2y.
5. R is the trapezoidal region determined by the lines y=−14x+52, y=0, y=2, and x = 0; \space \rho (x,y) = 3xy\).
- Answer
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a. Ix=88, Iy=1560, and I0=1648; b. Rx=√41819, Ry=√741010, and R0=2√195719
6. R is the trapezoidal region determined by the lines y=0, y=1, y=x, and y = -x + 3; \space \rho (x,y) = 2x + y\).
7. R is the disk of radius 2 centered at (1,2); ρ(x,y)=x2+y2−2x−4y+5.
- Answer
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a. Ix=128π3, Iy=56π3, and I0=184π3; b. Rx=4√33, Ry=√212, and R0=√693
8. R is the unit disk; ρ(x,y)=3x4+6x2y2+3y4.
9. R is the region enclosed by the ellipse x2+4y2=1; ρ(x,y)=1.
- Answer
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a. Ix=π32, Iy=π8, and I0=5π32; b. Rx=14, Ry=12, and R0=√54
10. R={(x,y)|9x2+y2≤1, x≥0, y≥0}; ρ(x,y)=√9x2+y2.
11. R is the region bounded by y=x, y=−x, y=x+2, and y=−x+2; ρ(x,y)=1.
- Answer
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a. Ix=73, Iy=13, and I0=83; b. Rx=√426, Ry=√66, and R0=2√33
12. R is the region bounded by y=1x, y=2x, y=1, and y=2; ρ(x,y)=4(x+y).
Exercise 4: Mass of a solid
1. Let Q be the solid unit cube. Find the mass of the solid if its density ρ is equal to the square of the distance of an arbitrary point of Q to the xy-plane.
- Answer
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m=13.
2. Let Q be the solid unit hemisphere. Find the mass of the solid if its density ρ is proportional to the distance of an arbitrary point of Q to the origin.
3. The solid Q of constant density 1 is situated inside the sphere x2+y2+z2=16 and outside the sphere x2+y2+z2=1. Show that the center of mass of the solid is not located within the solid.
4. Find the mass of the solid Q={(x,y,z)|1≤x2+z2≤25, y≤1−x2−z2} whose density is ρ(x,y,z)=k, where k>0.
5. [T] The solid Q={(x,y,z)|x2+y2≤9, 0≤z≤1, x≥0, y≥0} has density equal to the distance to the xy-plane. Use a CAS to answer the following questions.
a. Find the mass of Q.
b. Find the moments Mxy, Mxz and Myz about the xy-plane, xz-plane, and yz-plane, respectively.
c. Find the center of mass of Q.
d. Graph Q and locate its center of mass.
- Answer
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a. m=9π4; b. Mxy=3π2, Mxz=818, Myz=818; c. ˉx=92π, ˉy=92π, ˉz=23;
d.
6. Consider the solid Q={(x,y,z)|0≤x≤1, 0≤y≤2, 0≤z≤3} with the density function ρ(x,y,z)=x+y+1.
a. Find the mass of Q.
b. Find the moments Mxy, Mxz and Myz about the xy-plane, xz-plane, and yz-plane, respectively.
c. Find the center of mass of Q.
7. [T] The solid Q has the mass given by the triple integral ∫1−1∫π/40∫10r2dr dθ dz.
8. Use a CAS to answer the following questions.
Show that the center of mass of Q is located in the xy-plane. Graph Q and locate its center of mass.
ˉx=3√22π, ˉy=3(2−√2)2π, ˉz=0; 2. the solid Q and its center of mass are shown in the following figure.
10. The solid Q is bounded by the planes x+y+z=3, x=0, y=0, and z=0. Its density is ρ(x,y,z)=x+ay, where a>0. Show that the center of mass of the solid is located in the plane z=35 for any value of a.
11. Let Q be the solid situated outside the sphere x2+y2+z2=z and inside the upper hemisphere x2+y2+z2=R2, where R>1. If the density of the solid is ρ(x,y,z)=1√x2+y2+z2, find R such that the mass of the solid is 7π2.
12. The mass of a solid Q is given by ∫20∫√4−x20∫√16−x2−y2√x2+y2(x2+y2+z2)ndz dy dx, where n is an integer. Determine n such the mass of the solid is (2−√2)π.
- Answer
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n=−2
13. Let Q be the solid bounded above the cone x2+y2=z2 and below the sphere x2+y2+z2−4z=0. Its density is a constant k>0. Find k such that the center of mass of the solid is situated 7 units from the origin.
14. The solid Q={(x,y,z)|0≤x2+y2≤16, x≥0, y≥0, 0≤z≤x} has the density ρ(x,y,z)=k. Show that the moment Mxy about the xy-plane is half of the moment Myz about the yz-plane.
15. The solid Q is bounded by the cylinder x2+y2=a2, the paraboloid b2−z=x2+y2, and the xy-plane, where 0<a<b. Find the mass of the solid if its density is given by ρ(x,y,z)=√x2+y2.
16. Let Q be a solid of constant density k, where k>0, that is located in the first octant, inside the circular cone x^2 + y^2 = 9(z - 1)^2, and above the plane z = 0. Show that the moment M_{xy} about the xy-plane is the same as the moment M_{yz} about the xz-plane.
17. The solid Q has the mass given by the triple integral \int_0^1 \int_0^{\pi/2} \int_0^{r^3} (r^4 + r) \space dz \space d\theta \space dr.
b. Find the moment M_{xy} about the xy-plane.
18. The solid Q has the moment of inertia I_x about the yz-plane given by the triple integral \int_0^2 \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{\frac{1}{2}(x^2+y^2)}^{\sqrt{x^2+y^2}} (y^2 + z^2)(x^2 + y^2) dz \space dx \space dy.
a. Find the density of Q.
b. Find the moment of inertia I_z about the xy-plane.
- Answer
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a. \rho (x,y,z) = x^2 + y^2; b. \frac{16\pi}{7}
19. The solid Q has the mass given by the triple integral \int_0^{\pi/4} \int_0^{2 \space sec \space \theta} \int_0^1 (r^3 cos \space \theta \space sin \space \theta + 2r) dz \space dr \space d\theta.
a. Find the density of the solid in rectangular coordinates.
b. Find the moment M_{xz} about the xz-plane.
20. Let Q be the solid bounded by the xy-plane, the cylinder x^2 + y^2 = a^2, and the plane z = 1, where a > 1 is a real number. Find the moment M_{xy} of the solid about the xy-plane if its density given in cylindrical coordinates is \rho(x,y,z) = \frac{d^2f}{dr^2} (r), where f is a differentiable function with the first and second derivatives continuous and differentiable on (0,a).
- Answer
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M_{xy} = \pi (f(0) - f(a) + af'(a))
21. A solid Q has a volume given by \[\iint_D \int_a^b dA \space dz\), where D is the projection of the solid onto the xy-plane and a < b are real numbers, and its density does not depend on the variable z. Show that its center of mass lies in the plane z = \frac{a+b}{2}.
22. Consider the solid enclosed by the cylinder x^2 + z^2 = a^2 and the planes y = b and y = c, where a > 0 and b < c are real numbers. The density of Q is given by \rho(x,y,z) = f'(y), where f is a differential function whose derivative is continuous on (b,c). Show that if f(b) = f(c), then the moment of inertia about the xz-plane of Q is null.
23. [T] The average density of a solid Q is defined as \rho_{ave} = \frac{1}{V(Q)} \iiint_Q \rho(x,y,z) dV = \frac{m}{V(Q)}, where V(Q) and m are the volume and the mass of Q, respectively. If the density of the unit ball centered at the origin is \rho (x,y,z) = e^{-x^2-y^2-z^2}, use a CAS to find its average density. Round your answer to three decimal places.
23. Show that the moments of inertia I_x, \space I_y, and I_z about the yz-plane, xz-plane, and xy-plane, respectively, of the unit ball centered at the origin whose density is \rho (x,y,z) = e^{-x^2-y^2-z^2} are the same. Round your answer to two decimal places.
- Answer
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I_x = I_y = I_z \approx 0.84