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17calculus > integrals > surface area

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 xy-plane 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 straight-forward. The difficulty with this topic occurs when evaluating the integral, which can quickly become quite complicated. Consequently, most problems you get will be carefully hand-picked 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 x-axis

rotation about the y-axis

\(\int_{a}^{b}{2\pi y ~ ds}\)

\(\int_{c}^{d}{2\pi x ~ ds}\)

\(ds = \sqrt{1 + [f'(x)]^2}dx \) 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: \(y=\sqrt{x}\)
range: \(4\leq x\leq9\)
axis: x-axis

solution

Practice A02

equation: \(y=x^2-\frac{1}{8}\ln(x)\)
range: \(1\leq x\leq2\)
axis: y-axis

solution

Practice A03

equation: \(y=\sqrt{4-x^2}\)
range: \(-1\leq x\leq1\)
axis: x-axis

solution

Practice A04

equation: \(f(x)=(1/3)x^3\)
range: \([0,2]\)
axis: x-axis

solution

Practice A05

equation: \(f(x)=\sqrt[3]{x}\)
range: \([0,8]\)
axis: y-axis

solution

Practice A06

equation: \(y=\sqrt[3]{6x}\)
range: \(0 \leq x \leq 4/3\)
axis: y-axis

solution

Practice A07

equation: \(f(x)=\sqrt{x}\)
range: \(0\leq x\leq 4\)
axis: x-axis

answer

solution

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