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.
video by MIP4U 

rotation about the xaxis 

\(\int_{a}^{b}{2\pi y ~ ds}\) 
rotation about the yaxis 
\(\int_{c}^{d}{2\pi x ~ ds}\) 
\(ds = \sqrt{1 + [f'(x)]^2} ~dx \) 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.
video by PatrickJMT 

Practice
Unless otherwise instructed, calculate the surface area of the given line segment rotated about the given axis. Give all answers in exact form.
equation: \(y=\sqrt{x}\)  range: \(4\leq x\leq9\)  axis of rotation: xaxis 
Problem Statement 

Calculate the surface area of the line segment \(y=\sqrt{x}\); \(4\leq x\leq9\) rotated about the xaxis. Give your answer in exact form.
Solution 

He works this problem twice in two videos, using different \(ds\) equations.
video by PatrickJMT 

video by PatrickJMT 

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equation: \(y=x^2\frac{1}{8}\ln(x)\)  range: \(1\leq x\leq2\)  axis of rotation: yaxis 
Problem Statement 

Calculate the surface area of the line segment \(y=x^2\frac{1}{8}\ln(x)\); \(1\leq x\leq2\) rotated about the yaxis. Give your answer in exact form.
Solution 

video by Krista King Math 

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equation: \(y=\sqrt{4x^2}\)  range: \(1\leq x\leq1\)  axis of rotation: xaxis 
Problem Statement 

Calculate the surface area of the line segment \(y=\sqrt{4x^2}\); \(1\leq x\leq1\) rotated about the xaxis. Give your answer in exact form.
Solution 

video by Krista King Math 

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equation: \(f(x)=(1/3)x^3\)  range: \([0,2]\)  axis of rotation: xaxis 
Problem Statement 

Calculate the surface area of the line segment \(f(x)=(1/3)x^3\); \([0,2]\) rotated about the xaxis. Give your answer in exact form.
Solution 

video by MIP4U 

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equation: \(f(x)=\sqrt[3]{x}\)  range: \([0,8]\)  axis of rotation: yaxis 
Problem Statement 

Calculate the surface area of the line segment \(f(x)=\sqrt[3]{x}\); \([0,8]\) rotated about the yaxis. Give your answer in exact form.
Solution 

video by MIP4U 

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equation: \(y=\sqrt[3]{6x}\)  range: \(0 \leq x \leq 4/3\)  axis of rotation: yaxis 
Problem Statement 

Calculate the surface area of the line segment \(y=\sqrt[3]{6x}\); \(0 \leq x \leq 4/3\) rotated about the yaxis. Give your answer in exact form.
Solution 

video by MIP4U 

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equation: \(f(x)=\sqrt{x}\)  range: \(0\leq x\leq 4\)  axis of rotation: xaxis 
Problem Statement 

Calculate the surface area of the line segment \(f(x)=\sqrt{x}\); \(0\leq x\leq 4\) rotated about the xaxis. Give your answer in exact form.
Final Answer 

\([(17)^{3/2}1]\pi/6\)
Problem Statement 

Calculate the surface area of the line segment \(f(x)=\sqrt{x}\); \(0\leq x\leq 4\) rotated about the xaxis. Give your answer in exact form.
Solution 

video by Dr Chris Tisdell 

Final Answer 

\([(17)^{3/2}1]\pi/6\) 
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Practice Instructions
Unless otherwise instructed, calculate the surface area of the given line segment rotated about the given axis. Give all answers in exact form.