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 $[f(x<em>i) - g(x</em>i)]$.
    • Area of rectangles:
      $Area = \sum<em>{i=1}^{n} (f(x</em>i) - g(x_i)) \Delta x$
  • As $n \to \infty$:
    $Area = \lim<em>{n \to \infty} \sum</em>{i=1}^{n} (f(x<em>i) - g(x</em>i)) \Delta x = \int_a^b [f(x) - g(x)] \, dx$

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: $A = \frac{1}{2} (b<em>1 + b</em>2) h$

• 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.