Spherical Coordinates

Integral Review

Integration Basics

  • The integral under discussion involves variables x, y, and z.

  • A clever substitution may be necessary to perform the integral.

  • The three-dimensional domain of integration is part of a cylinder.

Sketching the Domain

  • The focus is on sketching the domain to visualize the integration limits.

  • Outer Limits:

    • The variable z is limited between 0 and 1.

    • The variable x ranges from -2 to 0.

Understanding the Solid

  • Observations on the solid being integrated:

    • For all possible x and z within the solid, when projected onto the xz-plane, the result is a rectangle of dimensions 2 by 1.

    • The volume integral is computed over this rectangular area while considering the height along the y-axis.

    • The domain of the surfaces is this rectangle.

Surface Definitions

  • Surface on left:

    • Defined by the equation y = 0.

    • Restricted to the range of x from -2 to 0.

  • Surface on right:

    • Defines the cylindrical shape of the volume.

  • The figure resembles a
    "cheese wedge" or a quarter piece of a cylindrical pipe.

Transitioning to Cylindrical Coordinates

  • Given the geometry, the integral can be set up in cylindrical coordinates.

  • Cylindrical Coordinates Setup:

    • Theta (θ) Range: Starts from θ = π/2 (positive y-axis) to θ = π (negative x-axis).

    • Radius (r) Range: r is between 0 and 2.

    • Z Range: z remains between 0 and 1.

Conversion Formulas

  • Variable Conversions:

    • x = r cos(θ)

    • y = r sin(θ)

    • z remains as z.

  • Remember to include the Jacobian (r) in integration.

Evaluating the Integral

  • Each of the limits of integration is a constant.

  • The integrand is in the form of a product of functions which leads to separation of the integrals:

    • Volumes can be split into products of individual integrals:

    • Integrate in the order dθ, dr, dz:

Specific Integrals

  1. Integrating dθ:

    • Focused on cos³(θ) and sin²(θ).

    • Apply the identity for odd/even powers of sine and cosine.

  2. Integrating dr:

    • The integrand leads to $r^6$.

  3. Integrating dz:

    • Simple integration of z.

Performing Exact Calculations

  • Each integral results in specific values:

    • The outcome was found to be (-\frac{128}{105}).

    • The negative sign indicates the integrand (x³y²z) is always negative in the defined domain since x < 0.

Physical Interpretation

  • While the integral provides a negative value, interpretation issues arise:

    • The volume is not computed; rather the function represents a density.

    • Possible interpretations include mass density under certain conditions.

Triple Integral Considerations

  • The need to break triple integrals into parts arises based on geometry.

  • Order of Integration: Understanding if dz can be innermost depends on the solid's top and bottom surfaces defined by functions of x and y.

Introduction to Spherical Coordinates

  • Spherical coordinates (ρ, θ, φ) define locations in three-dimensional space:

    • ρ: Distance from the origin (3D), NOT the 2D projection:

    • θ: Angle in the xy-plane.

    • φ: Angle down from the z-axis.

Coordinates Definition:

  • ρ must be >= 0, θ is in [0, 2π], and φ in [0, π]

  • Conversion from Cartesian to spherical coordinates involves formulas:

    • ( x = \rho \sin(\phi) \cos(\theta) )

    • ( y = \rho \sin(\phi) \sin(\theta) )

    • ( z = \rho \cos(\phi) )

Analyzing a Specific Example

  • Given specific spherical coordinates, translate into ρ, θ, and φ into Cartesian coordinates.

  • Example conversions necessitate sketching to visualize placement accurately.

  • The conversion equations allow calculation of exact x, y, and z values:

    • Examples given produce illustrative results confirming geometrical expectations.

Graphing in Spherical Coordinates

  • Simple spherical graphs include:

    • Constant ρ gives spheres.

    • Constant θ yields planes in specific orientations.

    • Constant φ leads to cones.

Volume Element in Spherical Coordinates

  • Volume differentials are expressed as:

    • ( dV = \rho^2 \sin(\phi) d\rho d\phi d\theta )

Finding Volumes Using Triple Integrals

  • Example integral inside a sphere defined by ρ from 0 to a results in straightforward calculations.

  • Another scenario included involves a cone and careful attention to integration limits which results in confirmed familiar outcomes for geometric processes.

Conclusion

  • Techniques for integrals are crucial in various coordinate systems, with spherical and cylindrical coordinates engaging different geometrical insights.

  • Visual aids enhance understanding of geometry and conversions, enriching the study of multidimensional integrals.