7: Parametric Equations and Polar Coordinates
- Page ID
- 130116
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- 7.1: Conic Sections
- Conic sections get their name because they can be generated by intersecting a plane with a cone. A cone has two identically shaped parts called nappes. Conic sections are generated by the intersection of a plane with a cone. If the plane is parallel to the axis of revolution (the y-axis), then the conic section is a hyperbola. If the plane is parallel to the generating line, the conic section is a parabola. If the plane is perpendicular to the axis of revolution, the conic section is a circle.
- 7.2: Parametric Equations
- In this section we examine parametric equations and their graphs. In the two-dimensional coordinate system, parametric equations are useful for describing curves that are not necessarily functions. The parameter is an independent variable that both x and y depend on, and as the parameter increases, the values of x and y trace out a path along a plane curve.
- 7.3: Calculus of Parametric Curves
- Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?
- 7.4: Polar Coordinates
- The rectangular coordinate system (or Cartesian plane) provides a means of mapping points to ordered pairs and ordered pairs to points. This is called a one-to-one mapping from points in the plane to ordered pairs. The polar coordinate system provides an alternative method of mapping points to ordered pairs. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates.
- 7.5: Area and Arc Length in Polar Coordinates
- In the rectangular coordinate system, the definite integral provides a way to calculate the area under a curve. In particular, if we have a function y=f(x) defined from x=a to x=b where f(x)>0 on this interval, the area between the curve and the x-axis is given by A=∫f(x)dx. This fact, along with the formula for evaluating this integral, is summarized in the Fundamental Theorem of Calculus. In this section, we study analogous formulas for area and arc length in the polar coordinate system.