Double Integrals Lecture Notes

Introduction to Double Integrals

• New topic: Double Integrals
• Purpose: Understanding their meaning and application.
• Connection to Calculus 1.

Review of Calculus 1

Key concepts:

1. Finding the Slope of the Tangent Line
   • Derivatives are used repeatedly in this context.
2. Finding the Area Under a Curve
   • Process:
     • Cut the interval [A, B] into smaller segments.
     • Use rectangles to approximate the area (Reiman sums).

Steps for Finding Area:

• Divide interval [A, B] into n equal segments:
  • Width of each rectangle:
    [ \Delta x = \frac{B - A}{n} ]
• Choose a point in each segment, denoted as X_K.
• Calculate the height of each rectangle by plugging the chosen x value into the function.
• Area of each rectangle:
  [ \text{Area} = \text{Width} \times \text{Height} ]
• Total Area: sum of areas of all rectangles as n approaches infinity for better approximation.

Transition to Double Integrals

• Single variable calculus gives area under curves.
• Double variable calculus extends this to volume under surfaces.

Concept of Volume Under a Surface:

1. Identify the region in the xy-plane bounded by [A, B] and [C, D].

2. Cut the rectangular region into equal segments:

   • n partitions in both x and y directions.

3. For each rectangle, pick a point to determine the height:

   • Denote the point as X_ij, Y_ij.

Calculating Volume:

• Volume of each rectangular prism = Surface Area × Height.
• Total Volume:
  • Calculate the sum of volumes for all rectangles.
  • Use double summation to account for both dimensions.

Transition to Limits:

• To move from finite to infinite sums:
  • Let both m and n approach infinity.
  • This allows us to use the concept of limits to find exact volumes.

Conclusion on Double Integrals

• Double Integrals: Represents the exact volume under a surface defined by a function over a specified region.
• Important notations:
  • [ \iint_{D} f(x, y) , dA ]
  • Shows integration over region D._

Examples and Applications

1. Example of Volume Approximation

   • Draw the region in the xy-plane.
   • Choose points to calculate volume under the surface.
   • Demonstrate the impact of different chosen points on the calculated volume.

2. Non-Rectangular Regions

   • Approach: Surround a non-rectangular area with rectangles.
   • If points lie within or on the boundary, include their values in the volume calculation.

Additional Topics

• Fubini's Theorem: Discusses the order of integration and how it affects results in non-rectangular regions.
• Importance of understanding the foundational concepts behind double integrals is emphasized.