- \n\t
- How much work is done when a person lifts a [latex]50[\/latex] lb box of comics onto a truck that is [latex]3[\/latex] ft off the ground?<\/li>\n\t
- Find the work done when you push a box along the floor [latex]2[\/latex] m, when you apply a constant force of [latex]F=100\\text{N}.[\/latex]<\/li>\n\t
- What is the work done moving a particle from [latex]x=0[\/latex] to [latex]x=1[\/latex] m if the force acting on it is [latex]F=3{x}^{2}[\/latex] N?<\/li>\n<\/ol>\n
For the following exercises (4-5), find the mass of the one-dimensional object.<\/strong><\/p>\n
- \n\t
- A car antenna that is [latex]3[\/latex] ft long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)=3x+2[\/latex] lb\/ft<\/li>\n\t
- A pencil that is [latex]4[\/latex] in. long (starting at [latex]x=2)[\/latex] and has a density function of [latex]\\rho (x)=5\\text{\/}x[\/latex] oz\/in.<\/li>\n<\/ol>\n
For exercises 6-8, find the mass of the two-dimensional object that is centered at the origin.<\/strong><\/p>\n
- \n\t
- An oversized hockey puck of radius [latex]2[\/latex] in. with density function [latex]\\rho (x)={x}^{3}-2x+5[\/latex]<\/li>\n\t
- A plate of radius [latex]10[\/latex] in. with density function [latex]\\rho (x)=1+ \\cos (\\pi x)[\/latex]<\/li>\n\t
- A disk of radius [latex]5[\/latex] cm with density function [latex]\\rho (x)=\\sqrt{3x}[\/latex]<\/li>\n<\/ol>\n
For the following exercises (9-18), solve each problem.<\/strong><\/p>\n
- \n\t
- A spring has a natural length of [latex]10[\/latex] cm. It takes [latex]2[\/latex] J to stretch the spring to [latex]15[\/latex] cm. How much work would it take to stretch the spring from [latex]15[\/latex] cm to [latex]20[\/latex] cm?<\/li>\n\t
- A spring requires [latex]5[\/latex] J to stretch the spring from [latex]8[\/latex] cm to [latex]12[\/latex] cm, and an additional [latex]4[\/latex] J to stretch the spring from [latex]12[\/latex] cm to [latex]14[\/latex] cm. What is the natural length of the spring?<\/li>\n\t
- A force of [latex]F=20x-{x}^{3}[\/latex] N stretches a nonlinear spring by [latex]x[\/latex] meters. What work is required to stretch the spring from [latex]x=0[\/latex] to [latex]x=2[\/latex] m?<\/li>\n\t
- For the cable in the preceding exercise, how much work is done to lift the cable [latex]50[\/latex] ft?<\/li>\n\t
- A pyramid of height [latex]500[\/latex] ft has a square base [latex]800[\/latex] ft by [latex]800[\/latex] ft. Find the area [latex]A[\/latex] at height [latex]h.[\/latex] If the rock used to build the pyramid weighs approximately [latex]w=100{\\text{lb\/ft}}^{3},[\/latex] how much work did it take to lift all the rock?<\/li>\n\t
- The force of gravity on a mass [latex]m[\/latex] is [latex]F=\\text{\u2212}((GMm)\\text{\/}{x}^{2})[\/latex] newtons. For a rocket of mass [latex]m=1000\\text{kg},[\/latex] compute the work to lift the rocket from [latex]x=6400[\/latex] to [latex]x=6500[\/latex] km. (Note<\/em>: [latex]G=6\u00d7{10}^{-17}{\\text{N m}}^{2}\\text{\/}{\\text{kg}}^{2}[\/latex] and [latex]M=6\u00d7{10}^{24}\\text{kg}\\text{.})[\/latex]<\/li>\n\t
- A rectangular dam is [latex]40[\/latex] ft high and [latex]60[\/latex] ft wide. Compute the total force [latex]F[\/latex] on the dam when
\n- \n\t
- the surface of the water is at the top of the dam and<\/li>\n\t
- the surface of the water is halfway down the dam.<\/li>\n<\/ol>\n<\/li>\n\t
- Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.<\/li>\n\t
- A cylinder of depth [latex]H[\/latex] and cross-sectional area [latex]A[\/latex] stands full of water at density [latex]\\rho .[\/latex] Compute the work to pump all the water to the top.<\/li>\n\t
- A cone-shaped tank has a cross-sectional area that increases with its depth: [latex]A=(\\pi {r}^{2}{h}^{2})\\text{\/}{H}^{3}.[\/latex] Show that the work to empty it is half the work for a cylinder with the same height and base.<\/li>\n<\/ol>\n
Moments and Centers of Mass<\/h2>\n
For the following exercises (1-3), calculate the center of mass for the collection of masses given.<\/strong><\/p>\n
- \n\t
- [latex]{m}_{1}=1[\/latex] at [latex]{x}_{1}=-1[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex]{x}_{2}=2[\/latex]<\/li>\n\t
- Unit masses at [latex](x,y)=(1,0),(0,1),(1,1)[\/latex]<\/li>\n\t
- [latex]{m}_{1}=1[\/latex] at [latex](1,0)[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex](2,2)[\/latex]<\/li>\n<\/ol>\n
For the following exercises (4-8), compute the center of mass [latex]\\overline{x}.[\/latex]<\/strong><\/p>\n
- \n\t
- [latex]\\rho ={x}^{2}[\/latex] for [latex]x\\in (0,L)[\/latex]<\/li>\n\t
- [latex]\\rho = \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/li>\n\t
- [latex]\\rho ={e}^{x}[\/latex] for [latex]x\\in (0,2)[\/latex]<\/li>\n\t
- [latex]\\rho =x \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/li>\n\t
- [latex]\\rho =\\text{ln}x[\/latex] for [latex]x\\in (1,e)[\/latex]<\/li>\n<\/ol>\n
For the following exercise, compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/strong><\/p>\n
- \n\t
- [latex]\\rho =3[\/latex] in the triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex]<\/li>\n<\/ol>\n
For the following exercises (10-13), use a calculator to draw the region, then compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/strong><\/p>\n
- \n\t
- The region bounded by [latex]y= \\cos (2x),[\/latex] [latex]x=-\\frac{\\pi }{4},[\/latex] and [latex]x=\\frac{\\pi }{4}[\/latex]<\/li>\n\t
- The region between [latex]y=\\frac{5}{4}{x}^{2}[\/latex] and [latex]y=5[\/latex]<\/li>\n\t
- The region bounded by [latex]y=0,[\/latex] [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/li>\n\t
- The region bounded by [latex]y={x}^{2}[\/latex] and [latex]y={x}^{4}[\/latex] in the first quadrant<\/li>\n<\/ol>\n
For the following exercises (14-15), use the theorem of Pappus to determine the volume of the shape.<\/strong><\/p>\n
- \n\t
- Rotating [latex]y=mx[\/latex] around the [latex]y[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/li>\n\t
- A general cylinder created by rotating a rectangle with vertices [latex](0,0),[\/latex] [latex](a,0),(0,b),[\/latex] and [latex](a,b)[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a cylinder?<\/li>\n<\/ol>\n
For the following exercises (16-18), use a calculator to draw the region enclosed by the curve. Find the area [latex]M[\/latex] and the centroid [latex](\\overline{x},\\overline{y})[\/latex] for the given shapes. Use symmetry to help locate the center of mass whenever possible.<\/strong><\/p>\n
- \n\t
- Quarter-circle: [latex]y=\\sqrt{1-{x}^{2}},[\/latex] [latex]y=0,[\/latex] and [latex]x=0[\/latex]<\/li>\n\t
- Lens: [latex]y={x}^{2}[\/latex] and [latex]y=x[\/latex]<\/li>\n\t
- Half-ring: [latex]{y}^{2}+{x}^{2}=1,[\/latex] [latex]{y}^{2}+{x}^{2}=4,[\/latex] and [latex]y=0[\/latex]<\/li>\n<\/ol>\n
For the following exercises (19-20), solve each problem.<\/strong><\/p>\n
- \n\t
- Find the generalized center of mass between [latex]y={a}^{2}-{x}^{2},[\/latex] [latex]x=0,[\/latex] and [latex]y=0.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/li>\n\t
- Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius [latex]a[\/latex] is positioned with the left end of the circle at [latex]x=b,[\/latex] [latex]b>0,[\/latex] and is rotated around the [latex]y[\/latex]-axis.\n\n
![\"This](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213317\/CNX_Calc_Figure_06_06_201.jpg\")
\n<\/span><\/p>\n<\/li>\n<\/ol>\n","rendered":"Physical Applications<\/h2>\n
For the following exercises (1-3), find the work done.<\/strong><\/p>\n
- \n
- How much work is done when a person lifts a [latex]50[\/latex] lb box of comics onto a truck that is [latex]3[\/latex] ft off the ground?<\/li>\n
- Find the work done when you push a box along the floor [latex]2[\/latex] m, when you apply a constant force of [latex]F=100\\text{N}.[\/latex]<\/li>\n
- What is the work done moving a particle from [latex]x=0[\/latex] to [latex]x=1[\/latex] m if the force acting on it is [latex]F=3{x}^{2}[\/latex] N?<\/li>\n<\/ol>\n
For the following exercises (4-5), find the mass of the one-dimensional object.<\/strong><\/p>\n
- \n
- A car antenna that is [latex]3[\/latex] ft long (starting at [latex]x=0)[\/latex] and has a density function of [latex]\\rho (x)=3x+2[\/latex] lb\/ft<\/li>\n
- A pencil that is [latex]4[\/latex] in. long (starting at [latex]x=2)[\/latex] and has a density function of [latex]\\rho (x)=5\\text{\/}x[\/latex] oz\/in.<\/li>\n<\/ol>\n
For exercises 6-8, find the mass of the two-dimensional object that is centered at the origin.<\/strong><\/p>\n
- \n
- An oversized hockey puck of radius [latex]2[\/latex] in. with density function [latex]\\rho (x)={x}^{3}-2x+5[\/latex]<\/li>\n
- A plate of radius [latex]10[\/latex] in. with density function [latex]\\rho (x)=1+ \\cos (\\pi x)[\/latex]<\/li>\n
- A disk of radius [latex]5[\/latex] cm with density function [latex]\\rho (x)=\\sqrt{3x}[\/latex]<\/li>\n<\/ol>\n
For the following exercises (9-18), solve each problem.<\/strong><\/p>\n
- \n
- A spring has a natural length of [latex]10[\/latex] cm. It takes [latex]2[\/latex] J to stretch the spring to [latex]15[\/latex] cm. How much work would it take to stretch the spring from [latex]15[\/latex] cm to [latex]20[\/latex] cm?<\/li>\n
- A spring requires [latex]5[\/latex] J to stretch the spring from [latex]8[\/latex] cm to [latex]12[\/latex] cm, and an additional [latex]4[\/latex] J to stretch the spring from [latex]12[\/latex] cm to [latex]14[\/latex] cm. What is the natural length of the spring?<\/li>\n
- A force of [latex]F=20x-{x}^{3}[\/latex] N stretches a nonlinear spring by [latex]x[\/latex] meters. What work is required to stretch the spring from [latex]x=0[\/latex] to [latex]x=2[\/latex] m?<\/li>\n
- For the cable in the preceding exercise, how much work is done to lift the cable [latex]50[\/latex] ft?<\/li>\n
- A pyramid of height [latex]500[\/latex] ft has a square base [latex]800[\/latex] ft by [latex]800[\/latex] ft. Find the area [latex]A[\/latex] at height [latex]h.[\/latex] If the rock used to build the pyramid weighs approximately [latex]w=100{\\text{lb\/ft}}^{3},[\/latex] how much work did it take to lift all the rock?<\/li>\n
- The force of gravity on a mass [latex]m[\/latex] is [latex]F=\\text{\u2212}((GMm)\\text{\/}{x}^{2})[\/latex] newtons. For a rocket of mass [latex]m=1000\\text{kg},[\/latex] compute the work to lift the rocket from [latex]x=6400[\/latex] to [latex]x=6500[\/latex] km. (Note<\/em>: [latex]G=6\u00d7{10}^{-17}{\\text{N m}}^{2}\\text{\/}{\\text{kg}}^{2}[\/latex] and [latex]M=6\u00d7{10}^{24}\\text{kg}\\text{.})[\/latex]<\/li>\n
- A rectangular dam is [latex]40[\/latex] ft high and [latex]60[\/latex] ft wide. Compute the total force [latex]F[\/latex] on the dam when\n
- \n
- the surface of the water is at the top of the dam and<\/li>\n
- the surface of the water is halfway down the dam.<\/li>\n<\/ol>\n<\/li>\n
- Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.<\/li>\n
- A cylinder of depth [latex]H[\/latex] and cross-sectional area [latex]A[\/latex] stands full of water at density [latex]\\rho .[\/latex] Compute the work to pump all the water to the top.<\/li>\n
- A cone-shaped tank has a cross-sectional area that increases with its depth: [latex]A=(\\pi {r}^{2}{h}^{2})\\text{\/}{H}^{3}.[\/latex] Show that the work to empty it is half the work for a cylinder with the same height and base.<\/li>\n<\/ol>\n
Moments and Centers of Mass<\/h2>\n
For the following exercises (1-3), calculate the center of mass for the collection of masses given.<\/strong><\/p>\n
- \n
- [latex]{m}_{1}=1[\/latex] at [latex]{x}_{1}=-1[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex]{x}_{2}=2[\/latex]<\/li>\n
- Unit masses at [latex](x,y)=(1,0),(0,1),(1,1)[\/latex]<\/li>\n
- [latex]{m}_{1}=1[\/latex] at [latex](1,0)[\/latex] and [latex]{m}_{2}=3[\/latex] at [latex](2,2)[\/latex]<\/li>\n<\/ol>\n
For the following exercises (4-8), compute the center of mass [latex]\\overline{x}.[\/latex]<\/strong><\/p>\n
- \n
- [latex]\\rho ={x}^{2}[\/latex] for [latex]x\\in (0,L)[\/latex]<\/li>\n
- [latex]\\rho = \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/li>\n
- [latex]\\rho ={e}^{x}[\/latex] for [latex]x\\in (0,2)[\/latex]<\/li>\n
- [latex]\\rho =x \\sin x[\/latex] for [latex]x\\in (0,\\pi )[\/latex]<\/li>\n
- [latex]\\rho =\\text{ln}x[\/latex] for [latex]x\\in (1,e)[\/latex]<\/li>\n<\/ol>\n
For the following exercise, compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/strong><\/p>\n
- \n
- [latex]\\rho =3[\/latex] in the triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex]<\/li>\n<\/ol>\n
For the following exercises (10-13), use a calculator to draw the region, then compute the center of mass [latex](\\overline{x},\\overline{y}).[\/latex] Use symmetry to help locate the center of mass whenever possible.<\/strong><\/p>\n
- \n
- The region bounded by [latex]y= \\cos (2x),[\/latex] [latex]x=-\\frac{\\pi }{4},[\/latex] and [latex]x=\\frac{\\pi }{4}[\/latex]<\/li>\n
- The region between [latex]y=\\frac{5}{4}{x}^{2}[\/latex] and [latex]y=5[\/latex]<\/li>\n
- The region bounded by [latex]y=0,[\/latex] [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{9}=1[\/latex]<\/li>\n
- The region bounded by [latex]y={x}^{2}[\/latex] and [latex]y={x}^{4}[\/latex] in the first quadrant<\/li>\n<\/ol>\n
For the following exercises (14-15), use the theorem of Pappus to determine the volume of the shape.<\/strong><\/p>\n
- \n
- Rotating [latex]y=mx[\/latex] around the [latex]y[\/latex]-axis between [latex]x=0[\/latex] and [latex]x=1[\/latex]<\/li>\n
- A general cylinder created by rotating a rectangle with vertices [latex](0,0),[\/latex] [latex](a,0),(0,b),[\/latex] and [latex](a,b)[\/latex] around the [latex]y[\/latex]-axis. Does your answer agree with the volume of a cylinder?<\/li>\n<\/ol>\n
For the following exercises (16-18), use a calculator to draw the region enclosed by the curve. Find the area [latex]M[\/latex] and the centroid [latex](\\overline{x},\\overline{y})[\/latex] for the given shapes. Use symmetry to help locate the center of mass whenever possible.<\/strong><\/p>\n
- \n
- Quarter-circle: [latex]y=\\sqrt{1-{x}^{2}},[\/latex] [latex]y=0,[\/latex] and [latex]x=0[\/latex]<\/li>\n
- Lens: [latex]y={x}^{2}[\/latex] and [latex]y=x[\/latex]<\/li>\n
- Half-ring: [latex]{y}^{2}+{x}^{2}=1,[\/latex] [latex]{y}^{2}+{x}^{2}=4,[\/latex] and [latex]y=0[\/latex]<\/li>\n<\/ol>\n
For the following exercises (19-20), solve each problem.<\/strong><\/p>\n
- \n
- Find the generalized center of mass between [latex]y={a}^{2}-{x}^{2},[\/latex] [latex]x=0,[\/latex] and [latex]y=0.[\/latex] Then, use the Pappus theorem to find the volume of the solid generated when revolving around the [latex]y[\/latex]-axis.<\/li>\n
- Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius [latex]a[\/latex] is positioned with the left end of the circle at [latex]x=b,[\/latex] [latex]b>0,[\/latex] and is rotated around the [latex]y[\/latex]-axis.\n
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- [latex]\\rho =3[\/latex] in the triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex]<\/li>\n<\/ol>\n
- A rectangular dam is [latex]40[\/latex] ft high and [latex]60[\/latex] ft wide. Compute the total force [latex]F[\/latex] on the dam when\n
- [latex]\\rho =3[\/latex] in the triangle with vertices [latex](0,0),[\/latex] [latex](a,0),[\/latex] and [latex](0,b)[\/latex]<\/li>\n<\/ol>\n
- A rectangular dam is [latex]40[\/latex] ft high and [latex]60[\/latex] ft wide. Compute the total force [latex]F[\/latex] on the dam when