### Key Concepts

## 1.1 Parametric Equations

- 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.

## 1.2 Calculus of Parametric Curves

- The derivative of the parametrically defined curve $x=x\left(t\right)$ and $y=y\left(t\right)$ can be calculated using the formula $\frac{dy}{dx}=\frac{{y}^{\prime}(t)}{{x}^{\prime}(t)}.$ 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 $A={\displaystyle {\int}_{{t}_{1}}^{{t}_{2}}y\left(t\right){x}^{\prime}\left(t\right)\phantom{\rule{0.2em}{0ex}}dt}.$ - The arc length of a parametric curve can be calculated by using the formula $s={\displaystyle {\int}_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt}.$
- The surface area of a volume of revolution revolved around the
*x*-axis is given by $S=2\pi {\displaystyle {\int}_{a}^{b}y\left(t\right)\sqrt{{\left({x}^{\prime}\left(t\right)\right)}^{2}+{\left({y}^{\prime}\left(t\right)\right)}^{2}}dt}.$ If the curve is revolved around the*y*-axis, then the formula is $S=2\pi {\displaystyle {\int}_{a}^{b}x\left(t\right)\sqrt{{\left({x}^{\prime}\left(t\right)\right)}^{2}+{\left({y}^{\prime}\left(t\right)\right)}^{2}}dt}.$

## 1.3 Polar Coordinates

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

$$x=r\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=r\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $$

and

$$r=\sqrt{{x}^{2}+{y}^{2}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\frac{y}{x}.$$ - 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.

## 1.4 Area and Arc Length in Polar Coordinates

- The area of a region in polar coordinates defined by the equation $r=f\left(\theta \right)$ with $\alpha \le \theta \le \beta $ is given by the integral $A=\frac{1}{2}{{\displaystyle {\int}_{\alpha}^{\beta}\left[f\left(\theta \right)\right]}}^{2}d\theta .$
- 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 $r=f\left(\theta \right)$ with $\alpha \le \theta \le \beta $ is given by the integral $L={\displaystyle {\int}_{\alpha}^{\beta}\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime}\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle {\int}_{\alpha}^{\beta}\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta}\right)}^{2}}d\theta}}.$

## 1.5 Conic Sections

- The equation of a vertical parabola in standard form with given focus and directrix is $y=\frac{1}{4p}{\left(x-h\right)}^{2}+k$ where
*p*is the distance from the vertex to the focus and $\left(h,k\right)$ are the coordinates of the vertex. - The equation of a horizontal ellipse in standard form is $\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$ where the center has coordinates $\left(h,k\right),$ the major axis has length 2
*a,*the minor axis has length 2*b*, and the coordinates of the foci are $\left(h\pm c,k\right),$ where ${c}^{2}={a}^{2}-{b}^{2}.$ - The equation of a horizontal hyperbola in standard form is $\frac{{\left(x-h\right)}^{2}}{{a}^{2}}-\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1$ where the center has coordinates $\left(h,k\right),$ the vertices are located at $\left(h\pm a,k\right),$ and the coordinates of the foci are $\left(h\pm c,k\right),$ 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 $r=\frac{ep}{1\pm e\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta}$ or $r=\frac{ep}{1\pm e\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta},$ where*p*represents the focal parameter. - To identify a conic generated by the equation $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,$ first calculate the discriminant $D=4AC-{B}^{2}.$ 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.