17calculus
Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Gradients
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > vector fields > surface integrals

Surface Integrals

on this page: ► what are surface integrals?     ► surface integrals of scalar functions     ► practice with scalar functions     ► surface orientation     ► surface integrals of vector fields     ► surface integrals - meaning and applications     ► practice with vector fields

There are several equations for surface integrals and which one you use depends on what form your equations are in. This is a similar situation that you encountered with line integrals.

There are two main groups of equations, ones for surface integrals of scalar-valued functions and a second group for surface integrals of vector fields (often called flux integrals). The following table places them side-by-side so that you can easily see the difference.

scalar-valued function

\(f(x,y,z)\)

vector field

\(\vec{F}(x,y,z)=M(x,y,z)\hat{i} + N(x,y,z)\hat{j} + P(x,y,z)\hat{k}\)

What Are Surface Integrals?

Before we get started with the details of surface integrals and how to evaluate them, let's watch a couple of great videos that will gently introduce you to surface integrals of scalar-valued functions and give you some examples. This is one of our favorite instructors and we think these videos are worth taking the time to watch.

Dr Chris Tisdell - Surface Integrals and Scalar-Valued Functions [2 videos 72min-15secs total]

Surface Integrals of Scalar-Valued Functions

The following equations are used when you are given a scalar-valued function over which you need to evaluate a surface integral. The form of the function is \(f(x,y,z)\). In general, your surface is parameterized as \(\vec{r}(u,v)=x(u,v)\hat{i} + y(u,v)\hat{j} + z(u,v)\hat{k}\). So to evaluate the integral of \(f(x,y,z)\) over the surface \(\vec{r}\), we use the equation

\( \iint\limits_S {f(x,y) ~ dS} = \iint\limits_R {f(x(u,v),y(u,v),z(u,v)) ~ \| \vec{r}_u \times \vec{r}_v \| ~ dA} \)

\(\vec{r}(u,v)\) is the parametric surface

R is the region in the uv-plane

\(\vec{r}_u\) and \(\vec{r}_v\) are the partial derivatives of \(\vec{r}\)

In the special case where we have the surface described as \(z=g(x,y)\), we can parameterize the surface as \(\vec{r}=x\hat{i}+y\hat{j}+g(x,y)\hat{k}\). This gives \( \| \vec{r}_x \times \vec{r}_y \| = \sqrt{1+[g_x]^2+[g_y]^2} \) and the surface integral can then be written as \( \iint\limits_S {f(x,y) ~ dS} = \iint\limits_R {f(x,y,z) ~ \sqrt{1+[g_x]^2+[g_y]^2} ~ dA} \) and R is the region in the xy-plane.

Okay, let's try some practice problems evaluating surface integrals of scalar functions.

Basic Problems

Practice 1

Evaluate \(\iint_S { x^2 y z ~ dS }\) where S is the part of the plane \(z=1+2x+3y\) that lies above the rectangle \(0 \leq x \leq 3, 0 \leq y \leq 2\).

answer

solution

Practice 2

Evaluate \(\iint_S {xy ~ dS}\) using a parametric surface where S is \(x^2+y^2=4, 0 \leq z \leq 8\) in the first octant.

answer

solution

Practice 3

Evaluate \(\iint_S {x^2+y^2}\) using a parametric surface where S is the hemisphere \(x^2+y^2+z^2=1\) above the xy-plane.

answer

solution

Practice 4

Integrate \(f(x,y,z)=xy\) over the surface \(z=4-2x-2y\) in the first octant.

answer

solution

Intermediate Problems

Practice 5

Compute the surface integral of \(\displaystyle{f(x,y,z)=\frac{2z^2}{x^2+y^2+z^2}}\) over the cap of the sphere \(x^2+y^2+z^2=9\), \(z \geq 2\).

answer

solution

Practice 6

A roof is given by the graph of \(g(x,y)=25+0.5x+0.5y\) over \(0\leq x\leq 40\), \(0\leq y\leq 20\). If the density of the roof is given by \(f(x,y,z)=150-2z\), determine the mass of the roof.

answer

solution

Surface Orientation

Before we discuss surface integrals over vector fields, we need to discuss surface orientation. Surface orientation is important because we need to know which direction the vector field is pointing, outside or inside, in order to determine the flux through the surface.

The two vectors that calculated above, \(\vec{r}_u\) and \(\vec{r}_v\) and tangent vectors to the surface. Using the cross product, we can calculate two possible normal vectors, \(\vec{N}_1 = \vec{r}_u \times \vec{r}_v\) and \(\vec{N}_2 = \vec{r}_v \times \vec{r}_u\). One vector points inward, the other points outward. We will use the first one, i.e. \(\vec{N}_1 = \vec{r}_u \times \vec{r}_v\) and divide by the length to get the unit vector \(\displaystyle{ \vec{N} = \frac{\vec{r}_u \times \vec{r}_v}{ \| \vec{r}_u \times \vec{r}_v \|} }\) which is called the upward unit normal.
[ Important Note: This may not always be the outward pointing normal but for our discussions we will work with surfaces where the outward pointing normal is this upward pointing normal. For your application, you will need to double check that you have the outward pointing normal. ]

Surface Integrals of Vector Fields

In this video, Dr Chris Tisdell continues his discussion of surface integrals and talks about vector fields. Again, this is a great video to watch.

Dr Chris Tisdell - Surface integrals + vector fields [25min]

Surface integrals over vector fields are often called flux integrals since we will often be calculating the flux through a closed surface. The flux of a vector field \(\vec{F}(x,y,z)=M(x,y,z)\hat{i}+N(x,y,z)\hat{j}+P(x,y,z)\hat{k}\) through a surface S with a unit normal vector \(\vec{N}\) is \( \iint\limits_S { \vec{F} \cdot \vec{N} ~ dS} \).
[ Note: In the above description, there are two N's. One is a function \(N(x,y,z)\) which is the j-component of the vector function. The other is \(\vec{N}\), a unit normal vector. They are distinct and unrelated and should not be confused. ]

When the surface is given in terms of \(z=g(x,y)\), we can calculate the unit normal vector as \(\displaystyle{ \frac{\nabla G}{\| \nabla G \|} }\). Since \(dS = \| \nabla G \| ~ dA \) where \(G(x,y,z)=z-g(x,y)\), the surface integral becomes \( \iint\limits_R { \vec{F} \cdot \nabla G ~ dA } \) where R is the projection of S in the xy-plane.

Surface Integrals - Meaning and Applications

The meaning of the surface integral depends on what the function \(f(x,y,z)\) or \(\vec{F}(x,y,z)\) represents. Here is a video clip giving some applications.

Evans Lawrence - Lecture 31 - Parametric Surfaces, Surface Integrals [29min-3sec]

In this final video, he gives more explanation of surface integrals and a couple of examples. He has a unique way of thinking about surface integrals and, as he says at the first of this video, surface integration is not easy. So it will help you to watch this video clip as well before going on to trying some on your own.

Evans Lawrence - Lecture 32 - More on Parametric Surfaces, Surface Integrals [23min-15sec]


Okay, you are now ready for some practice problems calculating surface integrals using vector functions.
Then you will be ready for the three dimensional versions of Green's Theorem, Stokes' Theorem and the Divergence Theorem.

Stokes' Theorem →
Divergence Theorem →

Basic Problems

Practice 7

Compute the flux of \(\vec{F}=\langle x,y,z \rangle\) across the surface \(z=4-x^2-y^2\), \(z \geq 0\) oriented up.

answer

solution

Practice 8

Determine the flux of \(\vec{F}=\langle 0,-1,-2\rangle\) across the surface \(z=6-x-y\) in the first octant. Use a downward orientation.

answer

solution

Practice 9

Determine the surface area of the cylinder given by \(\vec{r}=\langle 3\cos(u),3\sin(u),v\rangle\) for \(0 \leq u \leq 2\pi\), \(0 \leq v \leq 4\).

answer

solution

Practice 10

Determine the surface area of the sphere given by \(\vec{r}=\langle 2\sin(u)\cos(v), 2\sin(u)\sin(v), 2\cos(u)\rangle\) for \(0 \leq u \leq \pi \), \(0 \leq v \leq 2\pi \).

answer

solution

Intermediate Problems

Practice 11

Calculate the flux of \(\vec{F}=z\hat{i}+yz\hat{j}+2x\hat{k}\) across the upper hemisphere of the unit sphere oriented with outward-pointing normals.

answer

solution

Search 17Calculus

Real Time Web Analytics
menu top search
1
17
menu top search 17