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Applied Integration  Calculating Surface Area 

This page covers the topic of surface area of an explicitly defined smooth curve revolved around an axis in the xyplane in cartesian (rectangular) coordinates. [You can also calculate surface area in polar coordinates and for surfaces described parametrically.] 
Setting up the integral to calculate surface area is fairly straightforward. The difficulty with this topic occurs when evaluating the integral, which can quickly become quite complicated. Consequently, most problems you get will be carefully handpicked by your instructor or the textbook author so that you can evaluate the integrals with the techniques you know. The comments we made on the arc length page about tricks to evaluating these integrals apply here as well.
First, let's look at a video clip explaining how to derive the surface area equations.
MIP4U  Surface Area of Revolution  Part 1 of 2  
rotation about the xaxis 
rotation about the yaxis  
\(\int_{a}^{b}{2\pi y ~ ds}\) 
\(\int_{c}^{d}{2\pi x ~ ds}\)  
\(ds = \sqrt{1 + [f'(x)]^2}dx \) or \( ds = \sqrt{1 + [g'(y)]^2}dy \) 

Notes
1. \(ds\) in the last row above is the differential arc length as discussed on the arc length page. Using \(ds\) allows us to write the integral in a more compact form and it is easier to see what is going on.
2. Which \(ds\) you use depends on how the graph is described.
Here is a great video clip explaining these equations in more detail.
PatrickJMT  Finding Surface Area  Part 1  
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Practice Problems 

Instructions   Unless otherwise instructed, calculate the surface area of the given line segment rotated about the given axis. Give all answers in exact form.
Level A  Basic 
Practice A01  

equation: \(y=\sqrt{x}\)  
solution 
Practice A02  

equation: \(y=x^2\frac{1}{8}\ln(x)\)  
solution 
Practice A03  

equation: \(y=\sqrt{4x^2}\)  
solution 
Practice A04  

equation: \(f(x)=(1/3)x^3\)  
solution 
Practice A05  

equation: \(f(x)=\sqrt[3]{x}\)  
solution 
Practice A06  

equation: \(y=\sqrt[3]{6x}\)  
solution 