The Main Idea 

• Mass-Density Formula for One-Dimensional Objects:
  • For a thin rod along the [latex]x[/latex]-axis from [latex]a[/latex] to [latex]b[/latex]: [latex]m = \int_a^b \rho(x) dx[/latex]
• Mass-Density Formula for Two-Dimensional Disks:
  • For a disk of radius [latex]r[/latex] with radial density: [latex]m = \int_0^r 2\pi x \rho(x) dx[/latex]
• Concept of Density Functions:
  • Linear density: mass per unit length
  • Area density: mass per unit area
  • Radial density: density varying along the radius

Problem-Solving Strategies 

• One-Dimensional Case:
  • Partition the rod into segments
  • Approximate mass of each segment: [latex]m_i \approx \rho(x_i^*) \Delta x[/latex]
  • Sum and take the limit as [latex]n[/latex] approaches infinity
• Two-Dimensional Case:
  • Partition the disk into concentric washers
  • Approximate area of each washer: [latex]A_i \approx 2\pi x_i^* \Delta x[/latex]
  • Approximate mass of each washer: [latex]m_i \approx 2\pi x_i^* \rho(x_i^*) \Delta x[/latex]
  • Sum and take the limit as [latex]n[/latex] approaches infinity

The Main Idea 

• Definition of Work:
  • Work is the energy transfer when a force moves an object
  • Units: Joules (J); Force in Newtons (N), Distance in meters (m)
• Work Formula for Constant Force: [latex]W = F \cdot d[/latex] (Force × Distance)
• Work Formula for Variable Force: [latex]W = \int_a^b F(x) dx[/latex]
• Application to Springs:
  • Hooke’s Law: [latex]F(x) = kx[/latex]
  • [latex]k[/latex] is the spring constant (N/m)

Problem-Solving Strategy

• Partition the interval [latex][a,b][/latex] into [latex]n[/latex] segments
• Approximate work in each segment: [latex]W_i \approx F(x_i^*) \Delta x[/latex]
• Sum the work over all segments: [latex]W \approx \sum_{i=1}^n F(x_i^*) \Delta x[/latex]
• Take the limit as [latex]n \to \infty[/latex] to get the exact work: [latex]W = \lim_{n \to \infty} \sum_{i=1}^n F(x_i^*) \Delta x = \int_a^b F(x) dx[/latex]

The Main Idea 

• Work in Pumping:
  • Calculating work required to pump water out of tanks
  • Varies based on tank shape and water level
• Key Principle:
  • Work = Force (Weight of water) × Distance (Height lifted)
• General Work Formula: [latex]W = \int_a^b F(x) dx[/latex], where [latex]F(x)[/latex] depends on tank shape
• Weight-Density of Water:
  • [latex]9800[/latex] N/m³ (metric) or [latex]62.4[/latex] lb/ft³ (imperial)

Problem-Solving Strategy

1. Sketch the tank and choose a coordinate system
2. Calculate volume of a representative water layer
3. Determine force using weight-density
4. Calculate lifting distance for the layer
5. Estimate work for the layer: Force × Distance
6. Sum work for all layers (Riemann sum)
7. Take limit and evaluate integral for exact work

The Main Idea 

• Hydrostatic Pressure:
  • Pressure exerted by a fluid at rest due to its weight
  • [latex]p = \rho s[/latex], where [latex]\rho[/latex] is weight density and [latex]s[/latex] is depth
• Force on Horizontal Surface:
  • [latex]F = \rho As[/latex], where [latex]A[/latex] is area
• Force on Vertical or Angled Surface:
  • [latex]F = \int_a^b \rho w(x)s(x)dx[/latex]
  • [latex]w(x)[/latex]: width function
  • [latex]s(x)[/latex]: depth function
• Pascal’s Principle:
  • Pressure at a given depth is the same in all directions

Problem-Solving Strategy

1. Sketch the situation and choose a coordinate system
2. Determine depth [latex]s(x)[/latex] and width [latex]w(x)[/latex] functions
3. Identify the weight density [latex]\rho[/latex] of the liquid
4. Set up and evaluate the force integral: [latex]F = \int_a^b \rho w(x)s(x)dx[/latex]