Calculus Volume 3

# Key Concepts

Calculus Volume 3Key Concepts

### 1.1Parametric 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.

### 1.2Calculus of Parametric Curves

• The derivative of the parametrically defined curve $x=x(t)x=x(t)$ and $y=y(t)y=y(t)$ can be calculated using the formula $dydx=y′(t)x′(t).dydx=y′(t)x′(t).$ 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 $A=∫t1t2y(t)x′(t)dt.A=∫t1t2y(t)x′(t)dt.$
• The arc length of a parametric curve can be calculated by using the formula $s=∫t1t2(dxdt)2+(dydt)2dt.s=∫t1t2(dxdt)2+(dydt)2dt.$
• The surface area of a volume of revolution revolved around the x-axis is given by $S=2π∫aby(t)(x′(t))2+(y′(t))2dt.S=2π∫aby(t)(x′(t))2+(y′(t))2dt.$ If the curve is revolved around the y-axis, then the formula is $S=2π∫abx(t)(x′(t))2+(y′(t))2dt.S=2π∫abx(t)(x′(t))2+(y′(t))2dt.$

### 1.3Polar 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
$x=rcosθandy=rsinθx=rcosθandy=rsinθ$

and
$r=x2+y2andtanθ=yx.r=x2+y2andtanθ=yx.$
• 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.

### 1.4Area and Arc Length in Polar Coordinates

• The area of a region in polar coordinates defined by the equation $r=f(θ)r=f(θ)$ with $α≤θ≤βα≤θ≤β$ is given by the integral $A=12∫αβ[f(θ)]2dθ.A=12∫αβ[f(θ)]2dθ.$
• 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 $r=f(θ)r=f(θ)$ with $α≤θ≤βα≤θ≤β$ is given by the integral $L=∫αβ[f(θ)]2+[f′(θ)]2dθ=∫αβr2+(drdθ)2dθ.L=∫αβ[f(θ)]2+[f′(θ)]2dθ=∫αβr2+(drdθ)2dθ.$

### 1.5Conic Sections

• The equation of a vertical parabola in standard form with given focus and directrix is $y=14p(x−h)2+ky=14p(x−h)2+k$ where p is the distance from the vertex to the focus and $(h,k)(h,k)$ are the coordinates of the vertex.
• The equation of a horizontal ellipse in standard form is $(x−h)2a2+(y−k)2b2=1(x−h)2a2+(y−k)2b2=1$ where the center has coordinates $(h,k),(h,k),$ the major axis has length 2a, the minor axis has length 2b, and the coordinates of the foci are $(h±c,k),(h±c,k),$ where $c2=a2−b2.c2=a2−b2.$
• The equation of a horizontal hyperbola in standard form is $(x−h)2a2−(y−k)2b2=1(x−h)2a2−(y−k)2b2=1$ where the center has coordinates $(h,k),(h,k),$ the vertices are located at $(h±a,k),(h±a,k),$ and the coordinates of the foci are $(h±c,k),(h±c,k),$ where $c2=a2+b2.c2=a2+b2.$
• 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 $r=ep1±ecosθr=ep1±ecosθ$ or $r=ep1±esinθ,r=ep1±esinθ,$ where p represents the focal parameter.
• To identify a conic generated by the equation $Ax2+Bxy+Cy2+Dx+Ey+F=0,Ax2+Bxy+Cy2+Dx+Ey+F=0,$ first calculate the discriminant $D=4AC−B2.D=4AC−B2.$ If $D>0D>0$ then the conic is an ellipse, if $D=0D=0$ then the conic is a parabola, and if $D<0D<0$ then the conic is a hyperbola.
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