Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Calculus Volume 2

Key Concepts

Calculus Volume 2Key Concepts

Key Concepts

7.1 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.

7.2 Calculus 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=2πaby(t)(x(t))2+(y(t))2dt.S=2πaby(t)(x(t))2+(y(t))2dt. If the curve is revolved around the y-axis, then the formula is S=2πabx(t)(x(t))2+(y(t))2dt.S=2πabx(t)(x(t))2+(y(t))2dt.

7.3 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

  • 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.

7.4 Area 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 αθβαθβ is given by the integral A=12αβ[f(θ)]2dθ.A=12αβ[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 αθβαθβ 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.

7.5 Conic Sections

  • The equation of a vertical parabola in standard form with given focus and directrix is y=14p(xh)2+ky=14p(xh)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 (xh)2a2+(yk)2b2=1(xh)2a2+(yk)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=a2b2.c2=a2b2.
  • The equation of a horizontal hyperbola in standard form is (xh)2a2(yk)2b2=1(xh)2a2(yk)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=4ACB2.D=4ACB2. 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.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.


This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at
Citation information

© Feb 5, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.