Calculus II 10.03 Parametric Equations and Calculus
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Contents
10.3 Parametric Equations and Calculus
 Find the slope for a tangent line to a curve given by parametric equations.
 Find the arc length for a curve given by parametric equations.
 Find the area for a rotated surface in parametric form.
Slope and Tangent Lines

Consider the projectile represented by the parametric equations
as shown in Figure 10.3.1. The initial angle is \(45^{\circ}\) with slope \(m=\tan 45^{\circ} = 1 \). Can the slope be found at any time \(t\)? Theorem 10.3.1 provides a formula for finding the slope as a function for \(t\). 
Theorem 10.3.1 The Derivative for the Parametric Form

If a smooth curve \(C\) is given by the equations
then the slope for \(C\) at a point is
Proof Consider \(\Delta t >0\) and let
as shown in Figure 10.3.2.
Divide the numerator and denominator by \(\Delta t\) and differentiate for \(f\) and \(g\) to conclude.

Example 10.3.1 Differentiation and Parametric Form
Find \(dy/dx\) for the curve given by \(x=\sin t\) and \(y=\cos t\).
Solution
$$ \frac{dy}{dx} $$  $$= \frac{dy/dt}{dx/dt}$$ 
$$= \frac{\sin t}{\cos t} $$  
\(=  \tan t \) 
Because \(dy/dx\) is a function for \(t\), Theorem 10.3.1 can be used repeatedly to find higherorder derivatives. For example,
$$ \frac{d^{2}y}{dx^{2}} = \frac{d}{dx} \left [ \frac{d}{dx} \right ] = \frac{ \frac{d}{dt} \left [ \frac{dy}{dx} \right ] }{dx/dt}$$  Second derivative 
$$ \frac{d^{3}y}{dx^{3}} = \frac{d}{dx} \left [ \frac{d^{2}}{dx^{2}} \right ] = \frac{ \frac{d}{dt} \left [ \frac{d^{2}y}{dx^{2}} \right ] }{dx/dt} $$  Third derivative 
Example 10.3.2 Finding Slope and Concavity

For the curve given by
find the slope and concavity at the \((2,3)\).
the second derivative is
At \((2,3)\) \(t=4\) and the slope is
When \(t=4\) the second derivative is
The graph is concave upward at \((2,3)\) as shown in Figure 10.3.3. 
Example 10.3.3 A Curve with Two Tangent Lines at a Point

Because the parametric equations \(x=f(t)\) and \(y=g(t)\) need not define \(y\) as a function for \(x\), it is possible for a plane curve to loop around and cross itself. At these intersecting points, the curve may have more than one tangent line. The prolate cycloid[1] given by
crosses itself at \((0,2)\), as shown in Figure 10.3.4. Find the equations for both tangent lines at this point.
This reduces to, \(dy/dx=  \pi /2 \) when \(t=  \pi /2 \) and \(dy/dx= \pi /2 \) when \(t= \pi /2 \). Therefore the tangent lines are
The curve has a horizontal tangent at \((0,2\pi)\) when \(t=0\). A vertical tangent occurs at \((f(t_{0}),g(t_{0}))\) whenever \(dx/dt=0\) and \(dy/dt \ne0\), when \(t=t_{0}\). A horizontal tangent occurs at \((f(t_{0}),g(t_{0}))\) whenever \(dy/dt=0\) and \(dx/dt \ne 0\), when \(t=t_{0}\). 
Arc Length
How can the distance traveled by an object along parametric curve be determined? Recall from Section 7.4 that for a given curve \(C\) the formula for its arc length is given by \(y=h(x)\) over the interval \([x_{0},x_{1}]\) is
$$S$$  $$= \int_{x_{0}}^{x_{1}} \sqrt{ 1+ [h{}^{\prime}(x)]^{2} }\: dx $$ 
$$= \int_{x_{0}}^{x_{1}} \sqrt{ 1+ \left ( \frac{dy}{dx} \right)^{2} }\: dx $$  
If \(C\) is represented by the parametric equations \(x=f(t)\) and \(y=g(t)\), \(a \leqslant t \leqslant b\), and if \(dx/dt={f{}^{\prime}(t)} >0\), then the equation continues  
$$= \int_{x_{0}}^{x_{1}} \sqrt{ 1+ \left ( \frac{dy/dt}{dx/dt} \right)^{2} }\: dx $$  
$$= \int_{a}^{b} \sqrt{ \frac{(dx/dt)^{2}+(dy/dt)^{2}}{(dx/dt)^{2}} }\: \frac{dx}{dt}\: dt $$  
$$= \int_{a}^{b} \sqrt{ \left( \frac{dx}{dt}\right)^{2}+\left( \frac{dy}{dt}\right)^{2}}\: dt $$  
$$= \int_{a}^{b} \sqrt{[ f{}^{\prime}(t)]^{2}+[ g{}^{\prime}(t)]^{2}}\: dt. $$ 
Theorem 10.3.2 Arc Length in Parametric Form
If a smooth curve \(C\) is represented by the parametric equations \(x=f(t)\) and \(y=g(t)\) such that \(C\) doe snot intersect itself over the interval \(a \leqslant t \leqslant b\), except possibly at the endpoints, then the arc length for \(C\) over that interval is given by
 $$S = \int_{a}^{b} \sqrt{ \left( \frac{dx}{dt}\right)^{2}+\left( \frac{dy}{dt}\right)^{2}}\: dt = \int_{a}^{b} \sqrt{[ f{}^{\prime}(t)]^{2}+[ g{}^{\prime}(t)]^{2}}\: dt. $$
Example 10.3.4 Finding the Arc Length for a Epicycloid

An epicycloid[2] is traced by a point on the smaller circle as it rolls around the larger circle.
Find the distance traveled by the point in one complete trip about the larger circle.
For the epicycloid shown in Figure 10.3.5, an arc length 40 long seems about right because the circle's circumference with radius 6 is

Area for a Rotated Surface
The formula for a rotated surface area in rectangular form to develop a formula for a rotated surface area in parametric form.
Theorem 10.3.3 Area for a Rotated Surface in Parametric Form
If a smooth curve \(C\) given by \(x=f(t)\) and \(y=g(t)\) does not cross itself on an interval \(a \leqslant t \leqslant b\), then the area \(S\) for the rotated surface formed by revolving \(C\) about the coordinate axis is given by the following.
 $$1. \:S= 2 \pi\int_{a}^{b} g(t) \sqrt{ \left( \frac{dx}{dt}\right)^{2}+\left( \frac{dy}{dt}\right)^{2}}\: dt \:\:\:\: \color{red}{\text{Revolution about the }x\text{axis: }g(t) \geqslant 0} $$
 $$2. \:S= 2 \pi\int_{a}^{b} f(t) \sqrt{ \left( \frac{dx}{dt}\right)^{2}+\left( \frac{dy}{dt}\right)^{2}}\: dt \:\:\:\: \color{red}{\text{Revolution about the }y\text{axis: }f(t) \geqslant 0} $$
The differential form for the arc length is
 $$ds= \sqrt{ \left( \frac{dx}{dt}\right)^{2}+\left( \frac{dy}{dt}\right)^{2}}\: dt$$
Then the formulas are written as
 $$1. \:S= 2 \pi \int_{a}^{b} g(t)\: ds \:\:\:\: 2.\: S= 2 \pi \int_{a}^{b} f(t)\:ds $$
Example 10.3.5 Finding the Area for a Rotated Surface

Let \(C\) be the arc for the circle \(x^{2}+y^{2}=9\) between
as shown in Figure 10.3.6. Find the area for the surface formed by rotating \(C\) about the \(x\)axis.
The interval for \(t\) was determined by observing that \(t=0\) when \(x=3\) and \(t=\pi /3 \) when \(x=3/2\). On this interval, \(C\) is smooth and \(y\) is nonnegative, and Theorem 10.3.3 can be applied.

Internal Links
Parent Article: Calculus II 10 Conics Parametric Equations and Polar Coordinates