적분의 활용 전체 문제

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region between $y=2{x}^{2},$ $y=0,$ $x=0,$ and $x=1$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region between $y=\frac{5}{4}{x}^{2}$ and $y=5$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] Region between $y=\sqrt{x},$ $y=\ln (x),$ $x=1,$ and $x=4$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y=0,$ $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$

For the following exercises, use a calculator to draw the region, then compute the center of mass $\left(\overset{-}{x},\overset{-}{y}\right).$ Use symmetry to help locate the center of mass whenever possible.

[T] The region bounded by $y=0,$ $x=0,$ and $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$

[T] The region bounded by $y={x}^{2}$ and $y={x}^{4}$ in the first quadrant

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Quarter-circle: $y=\sqrt{1-{x}^{2}},$ $y=0,$ and $x=0$

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Triangle: $y=x,$ $y=2-x,$ and $y=0$

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Lens: $y={x}^{2}$ and $y=x$

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Ring: ${y}^{2}+{x}^{2}=1$ and ${y}^{2}+{x}^{2}=4$

For the following exercises, use a calculator to draw the region enclosed by the curve. Find the area $M$ and the centroid $\left(\overset{-}{x},\overset{-}{y}\right)$ for the given shapes. Use symmetry to help locate the center of mass whenever possible.

[T] Half-ring: ${y}^{2}+{x}^{2}=1,$ ${y}^{2}+{x}^{2}=4,$ and $y=0$

Find the center of mass $\left(\overset{-}{x},\overset{-}{y}\right)$ for a thin wire along the semicircle $y=\sqrt{1-{x}^{2}}$ with unit mass. (Hint: Use the theorem of Pappus.)

A 100 ft rope, weighing 0.1 lb/ft, hangs over the edge of a tall building.

(a) How much work is done pulling the entire rope to the top of the building? (b) How much rope is pulled in when half of the total work is done?

A 50 m rope, with a mass density of 0.2 kg/m, hangs over the edge of a tall building.

(a) How much work is done pulling the entire rope to the top of the building? (b) How much work is done pulling in the first 20 m?

A rope of length $\ell$ ft hangs over the edge of tall cliff. (Assume the cliff is taller than the length of the rope.) The rope has a weight density of $d$ lb/ft.

(a) How much work is done pulling the entire rope to the top of the cliff? (b) What percentage of the total work is done pulling in the first half of the rope? (c) How much rope is pulled in when half of the total work is done?

A 20 m rope with mass density of 0.5 kg/m hangs over the edge of a 10 m building. How much work is done pulling the rope to the top?

A crane lifts a 2,000 lb load vertically 30 ft with a 1” cable weighing 1.68 lb/ft.

(a) How much work is done lifting the cable alone? (b) How much work is done lifting the load alone? (c) Could one conclude that the work done lifting the cable is negligible compared to the work done lifting the load?

A box weighing 2 lb lifts 10 lb of sand vertically 50 ft. A crack in the box allows the sand to leak out such that 9 lb of sand is in the box at the end of the trip. Assume the sand leaked out at a uniform rate. What is the total work done in lifting the box and sand?

A force of 1000 lb compresses a spring 3 in. How much work is performed in compressing the spring?

A force of 2 N stretches a spring 5 cm. How much work is performed in stretching the spring?