## 17Calculus Polar Coordinates - Surface Area

##### 17Calculus

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.

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

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.