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Single Variable Calculus

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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 [5min-16secs]

video by MIP4U

rotation about the x-axis

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

rotation about the y-axis

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

\(ds = \sqrt{1 + [f'(x)]^2} ~dx \) 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 [1min-2secs]

video by PatrickJMT

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Surface Area - Practice Problems Conversion

[A01-1195] - [A02-1196] - [A03-1197] - [A04-1198] - [A05-1199] - [A06-1915] - [A07-2002]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

GOT IT. THANKS!

Instructions - - 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: x-axis

Problem Statement

equation: \(y=\sqrt{x}\)

range: \(4\leq x\leq9\)

axis of rotation: x-axis

Solution

He works this problem twice in two videos, using different ds equations.

1195 solution video

video by PatrickJMT

1195 solution video

video by PatrickJMT

close solution

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

range: \(1\leq x\leq2\)

axis of rotation: y-axis

Problem Statement

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

range: \(1\leq x\leq2\)

axis of rotation: y-axis

Solution

1196 solution video

video by Krista King Math

close solution

equation: \(y=\sqrt{4-x^2}\)

range: \(-1\leq x\leq1\)

axis of rotation: x-axis

Problem Statement

equation: \(y=\sqrt{4-x^2}\)

range: \(-1\leq x\leq1\)

axis of rotation: x-axis

Solution

1197 solution video

video by Krista King Math

close solution

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

range: \([0,2]\)

axis of rotation: x-axis

Problem Statement

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

range: \([0,2]\)

axis of rotation: x-axis

Solution

1198 solution video

video by MIP4U

close solution

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

range: \([0,8]\)

axis of rotation: y-axis

Problem Statement

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

range: \([0,8]\)

axis of rotation: y-axis

Solution

1199 solution video

video by MIP4U

close solution

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

range: \(0 \leq x \leq 4/3\)

axis of rotation: y-axis

Problem Statement

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

range: \(0 \leq x \leq 4/3\)

axis of rotation: y-axis

Solution

1915 solution video

video by MIP4U

close solution

equation: \(f(x)=\sqrt{x}\)

range: \(0\leq x\leq 4\)

axis of rotation: x-axis

Problem Statement

equation: \(f(x)=\sqrt{x}\)

range: \(0\leq x\leq 4\)

axis of rotation: x-axis

Final Answer

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

Problem Statement

equation: \(f(x)=\sqrt{x}\)

range: \(0\leq x\leq 4\)

axis of rotation: x-axis

Solution

2002 solution video

video by Dr Chris Tisdell

Final Answer

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

close solution
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