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17Calculus Polar Coordinates - Surface Area

17Calculus
Single Variable Calculus
Derivatives
Integrals
Multi-Variable Calculus
Precalculus
Functions

On this page we cover a common calculus problem involving polar coordinates, determining arc length.
As mentioned on the main polar coordinates page, polar coordinates are just parametric equations. If you are familiar with parametric equations, this material should be very intuitive.

To calculate the surface area of a polar curve revolved about an axis, we use these integrals.

The equation of the polar curve is in the form \( x=r(\theta) \cos(\theta) \) and \( y = r(\theta) \sin(\theta) \) and the curve that is being revolved is defined to be from \(\theta_0\) to \(\theta_1\). We also define \( ds = \sqrt{r^2 + [dr/d\theta]^2}~dt \).

surface area

rotation about the x-axis
(the polar axis)

\(\displaystyle{ S = 2\pi \int_{\theta_0}^{\theta_1}{y~ds} = }\) \(\displaystyle{ 2\pi \int_{\theta_0}^{\theta_1}{ r(\theta) \sin(\theta) \sqrt{r^2 + [dr/d\theta]^2}~dt } }\)

rotation about the y-axis

\(\displaystyle{ S = 2\pi \int_{\theta_0}^{\theta_1}{x~ds} = }\) \(\displaystyle{ 2\pi \int_{\theta_0}^{\theta_1}{ r(\theta) \cos(\theta) \sqrt{r^2 + [dr/d\theta]^2}~dt } }\)

So, why the \(ds\) term? If you look at the previous page on arc length, you will notice that this term appears in that integral. This is called a differential length and is just a convenient way of writing these integrals.

Calculus: Early Transcendental Functions 3rd Edition

Practice

Calculate the surface area formed by rotating the polar curve \(r=4\sin\theta\), \(0\leq\theta\leq\pi\) about the polar axis.

Problem Statement

Calculate the surface area formed by rotating the polar curve \(r=4\sin\theta\), \(0\leq\theta\leq\pi\) about the polar axis.

Solution

Here are two videos. The first one solves the given problem. The second one is almost identical except that the polar curve is \(r=\sin\theta\). It will help to watch both videos. Of course, the answer in the first video is four times the answer in the second.

Krista King Math - 1268 video solution

video by Krista King Math

MIP4U - 1268 video solution

video by MIP4U

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