Physical Applications of Integration: Get Stronger

Physical Applications

For the following exercises (1-3), find the work done.

  1. How much work is done when a person lifts a 5050 lb box of comics onto a truck that is 33 ft off the ground?
  2. Find the work done when you push a box along the floor 22 m, when you apply a constant force of F=100N.F=100N.
  3. What is the work done moving a particle from x=0x=0 to x=1x=1 m if the force acting on it is F=3x2F=3x2 N?

For the following exercises (4-5), find the mass of the one-dimensional object.

  1. A car antenna that is 33 ft long (starting at x=0)x=0) and has a density function of ρ(x)=3x+2ρ(x)=3x+2 lb/ft
  2. A pencil that is 44 in. long (starting at x=2)x=2) and has a density function of ρ(x)=5/xρ(x)=5/x oz/in.

For exercises 6-8, find the mass of the two-dimensional object that is centered at the origin.

  1. An oversized hockey puck of radius 22 in. with density function ρ(x)=x32x+5ρ(x)=x32x+5
  2. A plate of radius 1010 in. with density function ρ(x)=1+cos(πx)ρ(x)=1+cos(πx)
  3. A disk of radius 55 cm with density function ρ(x)=3xρ(x)=3x

For the following exercises (9-18), solve each problem.

  1. A spring has a natural length of 1010 cm. It takes 22 J to stretch the spring to 1515 cm. How much work would it take to stretch the spring from 1515 cm to 2020 cm?
  2. A spring requires 55 J to stretch the spring from 88 cm to 1212 cm, and an additional 44 J to stretch the spring from 1212 cm to 1414 cm. What is the natural length of the spring?
  3. A force of F=20xx3F=20xx3 N stretches a nonlinear spring by xx meters. What work is required to stretch the spring from x=0x=0 to x=2x=2 m?
  4. For the cable in the preceding exercise, how much work is done to lift the cable 5050 ft?
  5. A pyramid of height 500500 ft has a square base 800800 ft by 800800 ft. Find the area AA at height h.h. If the rock used to build the pyramid weighs approximately w=100lb/ft3,w=100lb/ft3, how much work did it take to lift all the rock?
  6. The force of gravity on a mass mm is F=((GMm)/x2)F=((GMm)/x2) newtons. For a rocket of mass m=1000kg,m=1000kg, compute the work to lift the rocket from x=6400x=6400 to x=6500x=6500 km. (Note: G=6×1017N m2/kg2G=6×1017N m2/kg2 and M=6×1024kg.)M=6×1024kg.)
  7. A rectangular dam is 4040 ft high and 6060 ft wide. Compute the total force FF on the dam when
    1. the surface of the water is at the top of the dam and
    2. the surface of the water is halfway down the dam.
  8. Find the work required to pump all the water out of the cylinder in the preceding exercise if the cylinder is only half full.
  9. A cylinder of depth HH and cross-sectional area AA stands full of water at density ρ.ρ. Compute the work to pump all the water to the top.
  10. A cone-shaped tank has a cross-sectional area that increases with its depth: A=(πr2h2)/H3.A=(πr2h2)/H3. Show that the work to empty it is half the work for a cylinder with the same height and base.

Moments and Centers of Mass

For the following exercises (1-3), calculate the center of mass for the collection of masses given.

  1. m1=1m1=1 at x1=1x1=1 and m2=3m2=3 at x2=2x2=2
  2. Unit masses at (x,y)=(1,0),(0,1),(1,1)(x,y)=(1,0),(0,1),(1,1)
  3. m1=1m1=1 at (1,0)(1,0) and m2=3m2=3 at (2,2)(2,2)

For the following exercises (4-8), compute the center of mass ¯x.¯¯¯x.

  1. ρ=x2 for x(0,L)
  2. ρ=sinx for x(0,π)
  3. ρ=ex for x(0,2)
  4. ρ=xsinx for x(0,π)
  5. ρ=lnx for x(1,e)

For the following exercise, compute the center of mass (¯x,¯y). Use symmetry to help locate the center of mass whenever possible.

  1. ρ=3 in the triangle with vertices (0,0), (a,0), and (0,b)

For the following exercises (10-13), use a calculator to draw the region, then compute the center of mass (¯x,¯y). Use symmetry to help locate the center of mass whenever possible.

  1. The region bounded by y=cos(2x), x=π4, and x=π4
  2. The region between y=54x2 and y=5
  3. The region bounded by y=0, x24+y29=1
  4. The region bounded by y=x2 and y=x4 in the first quadrant

For the following exercises (14-15), use the theorem of Pappus to determine the volume of the shape.

  1. Rotating y=mx around the y-axis between x=0 and x=1
  2. A general cylinder created by rotating a rectangle with vertices (0,0), (a,0),(0,b), and (a,b) around the y-axis. Does your answer agree with the volume of a cylinder?

For the following exercises (16-18), use a calculator to draw the region enclosed by the curve. Find the area M and the centroid (¯x,¯y) for the given shapes. Use symmetry to help locate the center of mass whenever possible.

  1. Quarter-circle: y=1x2, y=0, and x=0
  2. Lens: y=x2 and y=x
  3. Half-ring: y2+x2=1, y2+x2=4, and y=0

For the following exercises (19-20), solve each problem.

  1. Find the generalized center of mass between y=a2x2, x=0, and y=0. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis.
  2. Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius a is positioned with the left end of the circle at x=b, b>0, and is rotated around the y-axis.

    This figure is a torus. It has inner radius of b. Inside of the torus is a cross section that is a circle. The circle has radius a.