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You CAN Ace Calculus

17calculus > vector fields > surface integrals

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Surface Integrals

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 $$0 \leq x \leq 3, 0 \leq y \leq 2$$.

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.

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.

solution

Practice 4

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

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 $$x^2+y^2+z^2=9$$, $$z \geq 2$$.

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.

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.

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.

solution

Practice 9

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

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 $$0 \leq u \leq \pi$$, $$0 \leq v \leq 2\pi$$.

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.