Chapter Review

• Parametric equations \(x=x(t),\;y=y(t),\) where \(t\) is the parameter (p. 637)
• Plane curve \(C\): the collection of points \(( x, y) =(x(t) , y(t) ) \) (p. 637)
• Orientation: the direction a point on the curve defined for \(a\leq t \leq b\) moves as \(t\) moves from \(a\) to \(b\) (p. 637)
• Convert between parametric equations and rectangular equations (pp. 638-640)
• Cycloid: the curve represented by \(x(t) =a(t-\sin t),\;y(t) =a(1-\cos t)\) (p. 642)

• Smooth curve (p. 647)
• Slope of a tangent line to a smooth curve \(C\;\dfrac{{dy}}{{dx}}\, = \,\dfrac{{\dfrac{{dy}}{{dt}}}}{{\dfrac{{dx}}{{dt}}}} , \dfrac{dx}{dt}\neq 0\) (p. 648)
• Vertical tangent line to a smooth curve \(C\): \(\dfrac{dx}{dt}=0,\) but \(\dfrac{dy}{dt}\neq 0\) (p. 648)
• Horizontal tangent line to a smooth curve \(C\): \(\dfrac{dy}{dt}=0,\) but \(\dfrac{dx}{dt}\neq 0\) (p. 648)
• Arc length formula for parametric equations: \(s=\int_{a}^{b}\sqrt{\left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt}\right) ^{2}} dt\) (p. 651)

The surface area \(S\) of the solid of revolution generated by revolving a smooth curve \(C\) represented by the parametric equations \(x=x(t) ,\;y=y(t),\;a\leq t\leq b\)

• about the \(x\)-axis: \(S=2\pi \int_{a}^{b}y(t) \sqrt{\left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt}\right) ^{2}}\,dt\) (p. 657)
• about the \(y\)-axis: \(S=2\pi \int_{a}^{b}x(t) \sqrt{\left( \dfrac{dx}{dt}\right) ^{2}+\left( \dfrac{dy}{dt}\right) ^{2}}\,dt\) (p. 658)
• about the \(x\)-axis: if the smooth curve \(C\) is represented by a rectangular equation \(y = f(x)\): \(S=2\pi \int_{a}^{b}f( x) \sqrt{1+[f' ( x) ] ^{2}}\,dx\) (p. 659)

• Polar coordinates \(( r,\theta ) ;\) pole \(O\); polar axis (p. 661)
• A point \(P\) with polar coordinates \((r,\theta )\) also can be represented by \(( r,\theta +2n\pi ) \) or \(( -r,\theta +( 2n+1) \pi ) ,\;n\) an integer. (p. 663)
• The polar coordinates of the pole are \((0,\theta )\), where \(\theta \) is any angle. (p. 663)
• For polar coordinates \(( r,\theta ) \) and rectangular coordinates \(( x,y) \):

• \(x=r\cos \theta ,\;y=r\sin \theta \) (p. 664)
• \( r^{2}=x^{2}+y^{2}\) and \(\tan \theta = \dfrac{y}{x},\;x\neq 0\) (p. 665)
• \(r=y\) and \(\theta =\dfrac{\pi }{2}\) if \(x=0\) (p. 665)

• Table 1 gives polar equations for some lines and circles. (p. 668)

• Library of Polar Equations (Table 7) (p. 675)
• Arc length \(s\) of a curve represented by a polar equation from \(\theta =\alpha \) to \(\theta =\beta ,\;\alpha \leq \theta \leq \beta \): \(s=\int_{\alpha }^{\beta }\sqrt{\,r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}}\,d\theta \) (p. 675)

• The area \(A\) enclosed by the graph of \(r=f(\theta )\) and the rays \(\theta =\alpha \) and \(\theta =\beta \): \(A=\int_{\alpha }^{\beta }\dfrac{1}{2}r^{2} d\theta \) (p. 679)
• The surface area \(S\) of the solid of revolution obtained by revolving \(r=f( \theta) ,\;\alpha \leq \theta \leq \beta ,\;0\leq \alpha <\beta \leq 2\pi ,\) about the polar axis: \begin{eqnarray*} S&=&2\pi \int_{\alpha }^{\beta }f(\theta )\sin \theta \sqrt{[ f(\theta )] ^{2}+[ f' ( \theta) ] ^{2}} d\theta \\ &=&2\pi \int_{\alpha }^{\beta }r\sin \theta \sqrt{ r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}}d\theta \end{eqnarray*} (p. 682)

• The polar equation of a conic: Table 8 (p. 687)
• Eccentricity: Parabola: \(e=1;\) ellipse: \(e<1;\) hyperbola: \(e>1\) (p. 687)