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:

  • $ext{Integral} <br>ightarrow ext{limits of } heta ext{ and } r$

  • 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.