Welcome back, class! As we approach the end of Chapter 6, we are moving beyond geometry (Area and Volume) and applying our integration skills to physics and finance. This week, we are tackling Section 6.4 (Work) and Section 6.5 (Average Value of a Function).

Section 6.4: The Physics of Integration (Work)

In physics, we know that Work is defined as Force times Distance ($W = F \cdot d$) when the force is constant. However, calculus allows us to calculate work when the force changes over the distance. We use the definite integral:

$$W = \int_{a}^{b} F(x) \, dx$$

In the attached Group Work 6.4, we look at "The Weighty Chain." This is a classic problem involving lifting a bucket from a well. It challenges you to consider three layers of complexity:

  1. Constant Weight: Lifting just the 70 lb bucket (Force is constant).
  2. Variable Weight (Chain): The chain weighs $0.55$ lbs/ft. As you lift the bucket, the length of the hanging chain decreases, meaning the force required changes at every inch.
  3. Variable Weight (Leaking): The bucket has a hole! It starts at 70 lbs but loses weight at a constant rate until it hits 35 lbs at the top.

Use your integrals to sum up the work done over the 60-foot depth!

Section 6.5: Average Value of a Function

How do you find the average of a continuously changing value, like temperature or interest rates? We use the Average Value formula:

$$f_{ave} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx$$

In Group Work 6.5, titled "Choosing a Bank," you are asked to compare a simple interest rate of 6% against a variable rate modeled by a sine wave:

$$I(t) = 0.06 + 0.01 \sin \left( \frac{2\pi}{365}t \right)$$

By integrating this function over a time period ($t$), you can determine the total interest earned and calculate the average rate to see which bank offers the better deal.

Important Resources & Due Dates

Please review the PowerPoint slides for these sections to see the step-by-step derivation of these formulas.

  • Section 6-4 PowerPoint (Download above)
  • Section 6-5 PowerPoint (Download above)

Chapter 6 Take Home Test

We have now covered the necessary material for your assessment. The Chapter 6 Take Home Test is now available. This test covers the full scope of the chapter, including:

  • Section 6.1: Areas bounded by curves (like $y=2^x$ and $y=8$).
  • Section 6.2 & 6.3: Volumes of revolution (Rotating regions about the x-axis, y-axis, or lines like $y=-1$).
  • Section 6.4 & 6.5: Work and Average Value applications.

Reminder: The Take Home Test is due at the beginning of our next class on March 19th. Please show all your work, especially for the integration setups!