Volumes With Cross Sections

Area Formulas for Common Shapes: AP Calculus AB Rundown

This rundown covers area formulas for common geometric shapes, which are fundamental for understanding area calculations in calculus, particularly when dealing with integration and related rates problems.

1. Rectangle

• Formula:

  $$ A = lw $$
  where:

  • A is the area
  • l is the length
  • w is the width

2. Square

• Formula:

  $$ A = s^2 $$
  where:

  • A is the area
  • s is the side length

3. Triangle

• Formula:

  $$ A = \frac{1}{2}bh $$
  where:

  • A is the area
  • b is the base
  • h is the height (perpendicular to the base)

  Note on Height: The height must be perpendicular to the base. For non-right triangles, you may need to use trigonometry to find the height.

4. Circle

• Formula:

  $$ A = \pi r^2 $$
  where:

  • A is the area
  • r is the radius

5. Trapezoid

• Formula:

  $$ A = \frac{1}{2}(b_1 + b_2)h $$
  where:

  • A is the area
  • b_1 and b_2 are the lengths of the parallel bases
  • h is the height (perpendicular distance between the bases)

  Trapezoid Bases: Make sure you identify the parallel sides as the bases.

6. Parallelogram

• Formula:

  $$ A = bh $$
  where:

  • A is the area
  • b is the base
  • h is the height (perpendicular distance between the base and the opposite side)

  Parallelogram Height: The height is NOT the length of the slanted side.

7. Sector of a Circle

• Formula:

  $$ A = \frac{1}{2}r^2\theta $$
  where:

  • A is the area
  • r is the radius
  • \theta is the central angle in radians

  Radian Measure: Always ensure the angle is in radians when using this formula. Convert from degrees if necessary: radians = degrees * (π/180).

Application in Calculus

These formulas are crucial for:

• Area between curves: Finding the area between two functions, f(x) and g(x), between x = a and x = b:

  $$ A = \int_a^b |f(x) - g(x)| dx $$

• Volumes of revolution: Using methods like the disk/washer or shell method to find the volume of a solid generated by rotating a region around an axis. These methods rely on integrating the area of cross-sections (often circles or washers).

• Related rates problems: Setting up equations that relate the rates of change of different variables, often involving area formulas. For example, if the radius of a circle is increasing at a certain rate, you can use the area formula to find the rate at which the area is increasing.

• Optimization Problems: Where you might need to maximize or minimize an area given certain constraints.

Remember to always draw a diagram when possible and carefully identify the relevant dimensions and variables. Understanding these basic area formulas is essential for success in AP Calculus AB.