1.4: Arc Length of a Curve and Surface Area
- Page ID
- 163253
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Corequisite Topics
Corequisite Topics
The following (prerequisite) topics related to the material in this section are only to be covered in a class with corequisite support. Students in a class without corequisite support are assumed to have already mastered these topics.
- Evaluating integral functions and interpreting the meaning of such functions
- Using the Fundamental Theorem of Calculus, Part 1 (FTC1) to compute the derivative of a definite integral with variable limits
- Computing differentials
- Hyperbolic functions - definitions, identities, evaluating, and graphing
If you find yourself constantly needing to review these topics, then you might be better served in a Calculus II course with Corequisite Support.
- Determine the length of a curve, \(y=f(x)\), between two points.
- Determine the length of a curve, \(x=g(y)\), between two points.
- Find the surface area of a solid of revolution.
Arc Length of a Curve
Given the function \(f(x)=\sin(x)\), we can compute the area under the curve, and calculate the volume if rotated about a horizontal or vertical line. We now turn our attention to the actual length of the curve on a given interval. To motivate this, let’s derive a general formula.
Deriving the Formula
[Deriving the arc length time]
Let \( f(\lambda) \) be a smooth function over the \( \lambda \) interval \([a,b]\).1 Then the arc length, \( L \), of the graph of \( f(\lambda) \) over this interval is given by\[ L = \int_{\lambda = a}^{\lambda = b} \, ds, \nonumber \]where\[ s(k) = \int_{a}^{k} \sqrt{1 + \left[ f^{\prime}(t) \right]^2} \, dt \quad \left( \text{this is called the }\textbf{Arc Length Function} \right). \nonumber \]Therefore, \( ds \) can be written as\[ds = \sqrt{1+[f^{\prime}(x)]^2}\,dx \nonumber \]or\[ds = \sqrt{1+[f^{\prime}(y)]^2}\,dy, \nonumber \]depending on the given function and our needs.
1 This means \( f^{\prime} \) is continuous on \( \left[ a,b \right] \).
Because the arc length function always contains radicals, finding the arc length of a curve can be difficult (or impossible) to do analytically. Therefore, numerical integration techniques are often employed.
Computing Arc Lengths
Use Desmos to graph\[x^{2/3}+y^{2/3} = 1\nonumber \]and guess a value for the length of the curve. Finally, find the actual length of the curve.
Determine the length of\[ y= \left(\dfrac{3x}{2}\right)^{2/3}+1\nonumber \]on \(0 \leq x \leq \frac{2}{3} \left(3\right)^{3/2}\).
Find the length of the curve.\[ x = \dfrac{y^4}{8} + \dfrac{1}{4y^2}, \quad 1 \leq y \leq 5 \nonumber \]
Area of a Surface of Revolution
Given the function \(f(x)=\sin(x)\), we can compute the area under the curve, calculate the volume if rotated about a horizontal or vertical line, and find the arc length of the curve on a given interval. We now focus on the surface area if we revolved the function about a horizontal or vertical line.
Deriving the Formula
[Deriving the surface area time]
The surface area of the surface of revolution formed by revolving the graph of a smooth function about an axis is\[ S = \int_a^b 2 \pi r \, ds, \nonumber \]where \( r \) is the radius of rotation and\[ ds = \sqrt{1 + \left( f^{\prime}(x) \right)^2} \, dx \nonumber \]or\[ ds = \sqrt{1 + \left( g^{\prime}(y) \right)^2} \, dy. \nonumber \]
A great interpretation of the surface area integral\[ \int 2 \pi r \, ds \nonumber \]is that the surface area of the \( i^{\text{th}} \) band is the product of its circumference and its width. The circumference of the band is \( 2 \pi r_i \) and its width is \( \Delta s \).
Surface area integrals can often be evaluated using any combination of variables; however, try not to use a variable that will not make a function.
Since \(2 \pi r \, ds \geq 0\), we don’t have to worry about negative integrals as long as the limits of integration go from low to high.
Computing Surface Areas
Approximate the surface area of the solid obtained by rotating the given function about the given line.\[f(x)=\sin(x), \quad 0 \leq x \leq \pi, \quad \text{about the }x\text{-axis}\nonumber \]using \( R_4 \).
Determine the surface area of the solid obtained by rotating the given function about the given line.\[ y = \sqrt[3]{x}, \quad 1 \leq y \leq 2, \quad \text{about the }y\text{-axis}\nonumber \]