<th id="dahxx"></th>
      1. <form id="dahxx"></form>
        Paul's Online Notes
        Paul's Online Notes
        Home / Calculus II / Parametric Equations and Polar Coordinates / Surface Area with Polar Coordinates
        Show Mobile Notice Show All Notes Hide All Notes
        Mobile Notice
        You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

        Section 3-10 : Surface Area with Polar Coordinates

        We will be looking at surface area in polar coordinates in this section. Note however that all we’re going to do is give the formulas for the surface area since most of these integrals tend to be fairly difficult.

        We want to find the surface area of the region found by rotating,

        \[r = f\left( \theta \right)\hspace{0.25in}\hspace{0.25in}\alpha \le \theta \le \beta \]

        about the \(x\) or \(y\)-axis.

        As we did in the tangent and arc length sections we’ll write the curve in terms of a set of parametric equations.

        \[\begin{align*}x & = r\cos \theta \hspace{0.75in}y = r\sin \theta \\ & = f\left( \theta \right)\cos \theta \hspace{0.55in} = f\left( \theta \right)\sin \theta \end{align*}\]

        If we now use the parametric formula for finding the surface area we’ll get,

        \[\begin{align*}S & = \int{{2\pi y\,ds}}\hspace{0.5in}{\mbox{rotation about }}x - {\rm{axis}}\\ S & = \int{{2\pi x\,ds}}\hspace{0.5in}{\mbox{rotation about }}y - {\rm{axis}}\end{align*}\]

        where,

        \[ds = \sqrt {{r^2} + {{\left( {\frac{{dr}}{{d\theta }}} \right)}^2}} \,d\theta \hspace{0.25in}\hspace{0.25in}r = f\left( \theta \right),\,\,\,\,\,\alpha \le \theta \le \beta \]

        Note that because we will pick up a \(d\theta \) from the \(ds\) we’ll need to substitute one of the parametric equations in for \(x\) or \(y\) depending on the axis of rotation. This will often mean that the integrals will be somewhat unpleasant.

        老湿机影院,老湿机视频,老湿机影院视费,老湿机福利,成人视频