Polar Integrals

Understanding the Region for Integration

• Given: If x^2 = 2

  • Therefore, x = ext{root } 2

• Consider the interval:

  • Left endpoint: 0

  • Right endpoint: ext{root } 2

• Pick a favorite x value between 0 and ext{root } 2 for integration.

Defining the Integration Limits

• Integration starts at the curve defined by:

  • Lower curve: y = x^2

  • Upper curve: y = 2

• The differential dx accumulates vertical slices between these curves.

Double Integral Interpretation

• The expression integrated is x y^2

• One interpretation: This double integral represents the volume beneath the graph above the indicated region in the domain.

Integrating Functions

• A review of integrating vertical slices:

  • Integrate with respect to y while treating x as constant.

  • The anti-derivative of y is computed:

  • Integral of y: rac{y^3}{3}

  • Evaluate from y = 2 to y = x^2:

  • Results in: rac{(2)^3}{3} - rac{(x^2)^3}{3} = rac{8}{3} - rac{x^6}{3}

Function Volume Interpretation

• Recall the graph of the flat plane z = 1

  • This plane is 1 unit directly above the 2D region.

  • The integral represents the volume of the solid that has a height of 1 above the given 2D region.

Connection to Past Calculus Concepts

• Importance of recognizing calculations learned in prior calculus courses:

  • The notation remains consistent with the area calculations from Calculus II.

  • Process involves:

  • Integration from lower to upper limit: Top curve minus bottom curve.

Review of Trigonometric Integrals (Calc II Revisit)

• Three common integrals involving trig functions:

  1. First Integral:

  • Recommended method: Substitution.

  • Substitute u = heta^2

  • Resulting integral:

  • rac{1}{2} ext{cos}(u)

  • Result: rac{1}{2} ext{sin}( heta^2) + C

  1. Second Integral:

  • Use of trigonometric identity: rac{1 + ext{cos}(2 heta)}{2}

  • Split into two integrals.

  1. Third Integral:

  • Use substitution with u = ext{cos}( heta)

  • Resulting integral: - rac{1}{3}u^3 + C

The Importance of Polar Coordinates

• Need for polar coordinates arises when integrating over circular or symmetric regions.

  • Relation between rectangular coordinates and polar:

  • r^2 = x^2 + y^2

  • x = r ext{cos}( heta)

  • y = r ext{sin}( heta)

  • an( heta) = rac{y}{x}

Basics of Graphing in Polar Coordinates

• Knowledge of simple polar equations:

  • Example: r = ext{constant} forms a circle.

• More complex shapes have equations like r = ext{sin}(2 heta) which form roses or other patterns.

Double Integrals in Polar Coordinates

• Additional information on polar coordinates:

  • Area dA in polar is measured differently:

  • Area element: dA = r ext{dr} ext{d} heta

  • Changes incorporate the radial increase as we extend further outward.

Setting Up Double Integrals in Polar Coordinates

• Integral format:

  • ext{Integral}
    ightarrow ext{limits of } heta ext{ and } r

  • Example: Limits can change from constants to functions of heta.

• Ross laws for radius and angle dictate how to approach integrations in circular domains.

Examples for Double Integrals in Polar

• Example 1:

  • Create boundaries for theta and radius before proceeding.

  • Confirm integration strategies from previous lessons apply.

• Example 2:

  • Domain determination is crucial given circular boundaries must be acknowledged based on geometry, such as using equations for circles.

Conclusion and Key Reminders

• Always remember to use necessary substitutions and identities in order to compute integrals effectively.

• Approaching integrals in polar coordinates offers advantages when dealing with certain symmetrical shapes, making understanding their applications vital in calculus.