Chapter 10.2-10.3
Parametric Equations
When a parameter t is involved, x and y can be defined separately in terms of t, resulting in a parametric equation. Learn about parametric equation and their roles in calculus.
Chapter 10.2
Plane Curves and Parametric Equations
Definition of a Plane Curve:
If f and g are continuous functions of t on an interval I, then the equations
x = f(t)
and
y = f(t)
are called parametric equations and t is called the parameter. The set of points (x, y) obtained as t varies over the interval I is called the graph of the parametric equations. Taken together, the parametric equations and the graph are called a plane curve, denoted by C.
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NOTE: At times it is important to distinguish a graph (the set of points) and a curve (the points together with their defining parametric equations). When it is important, we will make the distinctions explicit. When it is not important, we will use C to represent the graph or the curve.
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By plotting the points in order of increasing values of t, the curve is traced out in a specific orientation. This is called the orientation of the curve.
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Eliminating the Parameter:
Finding the rectangular equation that represents the graph of a set of parametric equations.
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Steps:
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Take the parametric equations.
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Solve for t in one equation.
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Substitute into second equation.
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Simplify into rectangular equation.
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Definition of a Smooth Curve:
A curve C represented by x = f(t) and y = g(t) on an interval I is called smooth if f′ and g′ are continuous on I and not simultaneously 0, except possibly at the endpoints of I. The curve C is called piecewise smooth if it is smooth on each subinterval of some partition of I.
Chapter 10.3
Parametric Equations and Calculus
Theorem 10.7 - Parametric Form of the Derivative:
If a smooth curve C is given by the equations x = f(t) and y = g(t), then the slope of C at (x, y) is
dy/dx = (dy/dt)/(dx/dt),
dx/dt ≠ 0
Theorem 10.8 - Arc Length in Parametric Form:
If a smooth curve C is given by x = f(t) and y = g(t) such that C does not intersect itself on the interval a ≤ t ≤ b (except possibly at the endpoints), then the arc length of C over the interval is given by
s = ∫ (a -> b) √( [dx/dt]^2 + [dy/dt]^2 ) dt
∫ (a -> b) √( [f′(t)]^2 + [g′(t)]^2 ) dt
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If a circle rolls along a line, a point on its circumference will trace a path called a cycloid.
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If a circle rolls around the circumference of another circle, the path of the point is an epicycloid.
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Theorem 10.9 - Area of a Surface of Revolution:
If a smooth curve C given by x = f(t) and y = g(t) does not cross itself on an interval a ≤ t ≤ b, then the area S of the surface of revolution formed by revolving C about the coordinate axes is given by the following
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1. S = 2π ∫ (a -> b) g(t) √( [dx/dt]^2 + [dy/dt]^2 ) dt
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(revolution about the x-axis: g(t) ≥ 0)
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2. S = 2π ∫ (a -> b) f(t) √( [dx/dt]^2 + [dy/dt]^2 ) dt
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(revolution about the y-axis: f(t) ≥ 0)
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These formulas are easy to remember if you think of the differential arc length as
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ds = √( [dx/dt]^2 + [dy/dt]^2 ) dt
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Then the formulas are written as follows.
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1. S = 2π ∫ (a -> b) g(t) ds
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2. S = 2π ∫ (a -> b) f(t) ds
