Math Insight

Pages similar to: Examples of scalar line integrals

• Introduction to a line integral of a scalar-valued function
  Introduction with interactive graphics illustrating the line integral of a scalar-valued function and informally deriving the formula for calculating the integral from the parametrization of the curve.
• Line integrals are independent of parametrization
  Description of how the value of a line integral over a curve is independent of the parametrization of the curve.
• Introduction to a line integral of a vector field
  The concepts behind the line integral of a vector field along a curve are illustrated by interactive graphics representing the work done on a magnetic particle. The graphics motivate the formula for the line integral.
• The arc length of a parametrized curve
  Introduction to the arc length of a parametrized curve. The arc length definition is illustrated with interactive graphics.
• Alternate notation for vector line integrals
  An alternative notation for the line integral of a vector field is introduced.
• Line integrals as circulation
  Definition of circulation as the line integral of a vector field around a closed curve.
• Vector line integral examples
  Example of calculating line integrals of vector fields.
• The integrals of multivariable calculus
  A summary of the integrals of multivariable calculus, including calculation methods and their relationship to the fundamental theorems of vector calculus.
• Length of curves
  An integral to find the length of a curve.
• An introduction to parametrized curves
  An introduction to how a vector-valued function of a single variable can be viewed as parametrizing a curve. Interactive graphics illustrate the way in which the function maps an interval onto a curve.
• Derivatives of parameterized curves
  The derivative of the vector-valued function parameterizing a curve is shown to be a vector tangent to the curve.
• Tangent lines to parametrized curves
  The tangent vector given by the derivative of a parametrized curve forms the basis for the equation of a line tangent to the curve.
• Tangent line to parametrized curve examples
  Examples showing how to calculate the tangent line to a parameterized curve from the derivative of the underlying vector-valued function.
• Introduction to double integrals
  The definition of a double integral is motivated through a hair density example.
• Double integrals as iterated integrals
  A description of how to convert double integrals into two single integrals.
• Double integral examples
  Examples of integrating double integrals over rectangles and triangles.
• Double integrals as volume
  Explanation of how double integrals could be used to represent volume.
• Examples of changing the order of integration in double integrals
  Examples illustrating how to change the order of integration (or reverse the order of integration) in double integrals.
• Double integrals as area
  Explanation of how double integrals could be used to represent area.
• Double integrals where one integration order is easier
  Illustration of a region where integrating a double integral in one order is must easier than integrating in the other order.

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