You CAN Ace Calculus  

17calculus > vector fields > surface integrals  


Topics You Need To Understand For This Page
Calculus Main Topics
Single Variable Calculus 

MultiVariable Calculus 
Tools
math tools 

general learning tools 
additional tools 
Related Topics and Links
external links you may find helpful 

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 scalarvalued functions and a second group for surface integrals of vector fields (often called flux integrals). The following table places them sidebyside so that you can easily see the difference.
scalarvalued 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 scalarvalued 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 ScalarValued Functions [2 videos 72min15secs total]  
Surface Integrals of ScalarValued Functions 
The following equations are used when you are given a scalarvalued 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 uvplane 
\(\vec{r}_u\) 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 xyplane.
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 xyplane.  
answer 
solution 
Practice 4  

Integrate \(f(x,y,z)=xy\) over the surface \(z=42x2y\) 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 
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 jcomponent 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 \(G(x,y,z)=zg(x,y)\), the surface integral becomes \( \iint\limits_R { \vec{F} \cdot \nabla G ~ dA } \) where R is the projection of S in the xyplane.
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 [29min3sec]  
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 [23min15sec]  
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 \(z=4x^2y^2\), \(z \geq 0\) oriented up.  
answer 
solution 
Practice 8  

Determine the flux of \(\vec{F}=\langle 0,1,2\rangle\) across the surface \(z=6xy\) 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 