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 integration arc length

17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

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17calculus > integrals > 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 [5min-16secs]

video by MIP4U

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

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

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

 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

 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

 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

 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

 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

 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

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

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

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