Chapter 6 Calculus Notes on Areas, Volumes, and Wo&rk

6.1 Area of a Plane Region

  • Definition of Area:

    • Area is a non-negative number.
    • If the graph of $y = f(x)$ is below the x-axis, it results in negative area which cannot be considered valid.
    • However, it represents the negative of the area bounded by $y = f(x)$, $x = a$, $x = b$, and $y = 0$.

  • Example 1: Find the area bounded by the graph $y = \frac{1}{2} x^2 - 2$, the x-axis, $x = -1$, and $x = 2$.

  • Area Between Two Curves:

    • To find the area between two curves $f(x)$ and $g(x)$ where $g(x) < f(x)$ over the interval $[a, b]$:
    • The area can be found using:
      Area = \int_a^b (f(x) - g(x)) \, dx
    • Using Riemann sums, the area can be approximated as:
    • Partition $[a, b]$ into $n$ subintervals of width $\, \Delta x$ and height $[f(xi) - g(xi)]$.
      • Area of rectangles:
        Area = \sum{i=1}^{n} (f(xi) - g(x_i)) \Delta x
    • As $n \to \infty$:
      Area = \lim{n \to \infty} \sum{i=1}^{n} (f(xi) - g(xi)) \Delta x = \int_a^b [f(x) - g(x)] \, dx

6.2 Volumes by Cross Sections

Part 1: Area Formulas

  • Find the area formulas for different shapes in terms of s:

    • Square: $A = s^2$
    • Equilateral Triangle: $A = \frac{\sqrt{3}}{4} s^2$
    • Isosceles Right Triangle: $A = \frac{1}{2} s^2$
    • Semicircle: $A = \frac{1}{2} \pi r^2 \text{ with } r = \frac{s}{2} = \frac{\pi s^2}{8}$
    • Trapezoid: $A = \frac{1}{2} (b1 + b2) h$

  • Volume of Solids:

    • If the cross-sectional area of a solid along an axis is $A(x)$, volume is obtained by integrating:
    • If perpendicular to the x-axis:
      V = \int_a^b A(x) \, dx
    • If perpendicular to the y-axis:
      V = \int_c^d A(y) \, dy

Part 2: Volumes by Disks and Washers

  • Volume of Revolution:

    • When a region is revolved around a line.
    • Cross-sections are disks or washers depending on if there's a hole.
    • Volume of a disk:
      V = \pi r^2 h
    • For washers:
      • Around horizontal axis:
        V = \pi \int_a^b (R(x)^2 - r(x)^2) \, dx
      • Around vertical axis:
        V = \pi \int_c^d (R(y)^2 - r(y)^2) \, dy

  • Examples:

    • Example problems require setting up integrals for volumes generated by specific curves revolved around axes.

6.3 Volumes by Shells

  • Shell Method:
    • Useful for horizontal slices and sometimes simpler than the washer method.
    • Volume of cylindrical shells:
    • V = 2 \pi \int_a^b (radius)(height) \, dx

6.4 Arc Length

  • Finding Curve Length:
    • Length can be approximated with line segments.
    • If $f'$ is continuous on $[a, b]$:
    • L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx
    • For curves stated as $x = g(y)$:
    • L = \int_c^d \sqrt{1 + (g'(y))^2} \, dy

6.5 Work

  • Work Formula:
    • $W = (force)(distance)$
  • Variable Force:
    • If force is variable, integrate to find work over a distance.
    • Example problems include finding work with Hooke's law and in pumping liquids.