Ch. 1 Key Concepts - Calculus Volume 3

Parametric equations provide a convenient way to describe a curve. A parameter can represent time or some other meaningful quantity.
It is often possible to eliminate the parameter in a parameterized curve to obtain a function or relation describing that curve.
There is always more than one way to parameterize a curve.
Parametric equations can describe complicated curves that are difficult or perhaps impossible to describe using rectangular coordinates.

The derivative of the parametrically defined curve $x = x (t)$ and $y = y (t)$ can be calculated using the formula $\frac{d y}{d x} = \frac{y^{'} (t)}{x^{'} (t)} .$ Using the derivative, we can find the equation of a tangent line to a parametric curve.
The area between a parametric curve and the x-axis can be determined by using the formula $A = \int_{t_{1}}^{t_{2}} y (t) x^{'} (t) d t .$
The arc length of a parametric curve can be calculated by using the formula $s = \int_{t_{1}}^{t_{2}} \sqrt{{(\frac{d x}{d t})}^{2} + {(\frac{d y}{d t})}^{2}} d t .$
The surface area of a volume of revolution revolved around the x-axis is given by $S = 2 π \int_{a}^{b} y (t) \sqrt{{(x^{'} (t))}^{2} + {(y^{'} (t))}^{2}} d t .$ If the curve is revolved around the y-axis, then the formula is $S = 2 π \int_{a}^{b} x (t) \sqrt{{(x^{'} (t))}^{2} + {(y^{'} (t))}^{2}} d t .$

The polar coordinate system provides an alternative way to locate points in the plane.
Convert points between rectangular and polar coordinates using the formulas

$x = r cos θ and y = r sin θ$

and

$r = \sqrt{x^{2} + y^{2}} and tan θ = \frac{y}{x} .$
To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.
Use the conversion formulas to convert equations between rectangular and polar coordinates.
Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.

The area of a region in polar coordinates defined by the equation $r = f (θ)$ with $α \leq θ \leq β$ is given by the integral $A = \frac{1}{2} {\int_{α}^{β} [f (θ)]}^{2} d θ .$
To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
The arc length of a polar curve defined by the equation $r = f (θ)$ with $α \leq θ \leq β$ is given by the integral $L = \int_{α}^{β} \sqrt{{[f (θ)]}^{2} + {[f^{'} (θ)]}^{2}} d θ = \int_{α}^{β} \sqrt{r^{2} + {(\frac{d r}{d θ})}^{2}} d θ .$

The equation of a vertical parabola in standard form with given focus and directrix is $y = \frac{1}{4 p} {(x - h)}^{2} + k$ where p is the distance from the vertex to the focus and $(h, k)$ are the coordinates of the vertex.
The equation of a horizontal ellipse in standard form is $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$ where the center has coordinates $(h, k),$ the major axis has length 2a, the minor axis has length 2b, and the coordinates of the foci are $(h \pm c, k),$ where $c^{2} = a^{2} - b^{2} .$
The equation of a horizontal hyperbola in standard form is $\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$ where the center has coordinates $(h, k),$ the vertices are located at $(h \pm a, k),$ and the coordinates of the foci are $(h \pm c, k),$ where $c^{2} = a^{2} + b^{2} .$
The eccentricity of an ellipse is less than 1, the eccentricity of a parabola is equal to 1, and the eccentricity of a hyperbola is greater than 1. The eccentricity of a circle is 0.
The polar equation of a conic section with eccentricity e is $r = \frac{e p}{1 \pm e cos θ}$ or $r = \frac{e p}{1 \pm e sin θ},$ where p represents the focal parameter.
To identify a conic generated by the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0,$ first calculate the discriminant $D = 4 A C - B^{2} .$ If $D > 0$ then the conic is an ellipse, if $D = 0$ then the conic is a parabola, and if $D < 0$ then the conic is a hyperbola.