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Section 16.5 Parametrized Surfaces and Surface Area

Thus far we have focused mostly on 2-dimensional vector fields, measuring flow and flux along/across curves in the plane. Both Green's Theorem and the Divergence Theorem make connections between planar regions and their boundaries. We now move our attention to 3-dimensional vector fields, considering both curves and surfaces in space.

Subsection 16.5.1 Parametrizing surfaces

Figure 16.5.1. Introducing parametrized surfaces

We are accustomed to describing surfaces as functions of two variables, usually written as \(z=f(x,y)\text{.}\) For our coming needs, this method of describing surfaces will prove to be insufficient. Instead, we will parametrize our surfaces, describing them as the set of terminal points of some vector-valued function \(\vec r(u,v) =\langle f(u,v),g(u,v),h(u,v)\rangle\text{.}\) The bulk of this section is spent practicing the skill of describing a surface \(\surfaceS\)using a vector-valued function. Once this skill is developed, we'll show how to find the surface area \(S\) of a parametrically-defined surface \(\surfaceS\text{,}\) a skill needed in the remaining sections of this chapter.

Definition 16.5.2. Parametrized Surface.

Let \(\vec r(u,v) = \langle\, f(u,v),g(u,v),h(u,v)\rangle\) be a vector-valued function that is continuous and one to one on the interior of its domain \(R\) in the \(u\)-\(v\) plane. The set of all terminal points of \(\vec r\) (i.e., the range of \(\vec r\) ) is the surface \(\surfaceS\text{,}\) and \(\vec r\) along with its domain \(R\) form a parametrization of \(\surfaceS\text{.}\)

This parametrization is smooth on \(R\) if \(\vec r_u\) and \(\vec r_v\) are continuous and \(\vec r_u\times \vec r_v\) is never \(\vec 0\) on the interior of \(R\text{.}\)

Given a point \((u_0,v_0)\) in the domain of a vector-valued function \(\vec r\text{,}\) the vectors \(\vec r_u(u_0,v_0)\) and \(\vec r_v(u_0,v_0)\) are tangent to the surface \(\surfaceS\) at \(\vec r(u_0,v_0)\) (a proof of this is developed later in this section). The definition of smoothness dictates that \(\vec r_u\times \vec r_v \neq \vec 0\text{;}\) this ensures that neither \(\vec r_u\) nor \(\vec r_v\) are \(\vec 0\text{,}\) nor are they ever parallel. Therefore smoothness guarantees that \(\vec r_u\) and \(\vec r_v\) determine a plane that is tangent to \(\surfaceS\text{.}\)

A surface \(\surfaceS\) is said to be orientable if a field of normal vectors can be defined on \(\surfaceS\) that vary continuously along \(\surfaceS\text{.}\) This definition may be hard to understand; it may help to know that orientable surfaces are often called “two sided.” A sphere is an orientable surface, and one can easily envision an “inside” and “outside” of the sphere. A paraboloid is orientable, where again one can generally envision “inside” and “outside” sides (or “top” and “bottom” sides) to this surface. Just about every surface that one can imagine is orientable, and we'll assume all surfaces we deal with in this text are orientable.

It is enlightening to examine a classic non-orientable surface: the Möbius band, shown in Figure 16.5.3. Vectors normal to the surface are given, starting at the point indicated in the figure. These normal vectors “vary continuously” as they move along the surface. Letting each vector indicate the “top” side of the band, we can easily see near any vector which side is the “top”.

However, if as we progress along the band, we recognize that we are labeling “both sides” of the band as the top; in fact, there are not two “sides” to this band, but one. The Möbius band is a non-orientable surface.

Figure 16.5.3. A Möbius band, a non-orientable surface

We now practice parametrizing surfaces.

Example 16.5.4. Parametrizing a surface over a rectangle.

Parametrize the surface \(z=x^2+2y^2\) over the rectangular region \(R\) defined by \(-3\leq x\leq 3\text{,}\) \(-1\leq y\leq 1\text{.}\)

Solution

There is a straightforward way to parametrize a surface of the form \(z=f(x,y)\) over a rectangular domain. We let \(x=u\) and \(y=v\text{,}\) and let \(\vec r(u,v) = \langle u,v, f(u,v)\rangle\text{.}\) In this instance, we have \(\vec r(u,v) = \langle u,v,u^2+2v^2\rangle\text{,}\) for \(-3\leq u\leq 3\text{,}\) \(-1\leq v\leq 1\text{.}\) This surface is graphed in Figure 16.5.5.

Figure 16.5.5. The surface parametrized in Example 16.5.4
Example 16.5.6. Parametrizing a surface over a circular disk.

Parametrize the surface \(z=x^2+2y^2\) over the circular region \(R\) enclosed by the circle of radius 2 that is centered at the origin.

Solution

We can parametrize the circular boundary of \(R\) with the vector-valued function \(\la 2\cos u,2\sin u\ra\text{,}\) where \(0\leq u\leq 2\pi\text{.}\) We can obtain the interior of \(R\) by scaling this function by a variable amount, i.e., by multiplying by \(v\text{:}\) \(\la 2v\cos u,2v\sin u\ra\text{,}\) where \(0\leq v\leq 1\text{.}\)

It is important to understand the role of \(v\) in the above function. When \(v=1\text{,}\) we get the boundary of \(R\text{,}\) a circle of radius 2. When \(v=0\text{,}\) we simply get the point \((0,0)\text{,}\) the center of \(R\) (which can be thought of as a circle with radius of 0). When \(v=1/2\text{,}\) we get the circle of radius \(1\) that is centered at the origin, which is the circle halfway between the boundary and the center. As \(v\) varies from 0 to 1, we create a series of concentric circles that fill out all of \(R\text{.}\)

Figure 16.5.7. The surface parametrized in Example 16.5.6

Thus far, we have determined the \(x\) and \(y\) components of our parametrization of the surface: \(x=2v\cos u\) and \(y=2v\sin u\text{.}\) We find the \(z\) component simply by using \(z = f(x,y) = x^2+2y^2\text{:}\)

\begin{equation*} z = (2v\cos u)^2+2(2v\sin u)^2 = 4v^2\cos^2u+8v^2\sin^2u\text{.} \end{equation*}

Thus \(\vec r(u,v) = \langle 2v\cos u,2v\sin u,4v^2\cos^2u+8v^2\sin^2u\rangle\text{,}\) \(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{,}\) which is graphed in Figure 16.5.7. The way that this graphic was generated highlights how the surface was parametrized. When viewing from above, one can see lines emanating from the origin; they represent different values of \(u\) as \(u\) sweeps from an angle of 0 up to \(2\pi\text{.}\) One can also see concentric circles, each corresponding to a different value of \(v\text{.}\)

Examples 16.5.4 and Example 16.5.6 demonstrate an important principle when parametrizing surfaces given in the form \(z=f(x,y)\) over a region \(R\text{:}\) if one can determine \(x\) and \(y\) in terms of \(u\) and \(v\text{,}\) then \(z\) follows directly as \(z=f(x,y)\text{.}\)

In the following two examples, we parametrize the same surface over triangular regions. Each will use \(v\) as a “scaling factor” as done in Example 16.5.6.

Example 16.5.8. Parametrizing a surface over a triangle.

Parametrize the surface \(z=x^2+2y^2\) over the triangular region \(R\) enclosed by the coordinate axes and the line \(y=2-2x/3\text{,}\) as shown in Figure 16.5.9.(a).

(a)
(b)
Figure 16.5.9. Part (a) shows a graph of the region \(R\text{,}\) and part (b) shows the surface over \(R\text{,}\) as defined in Example 16.5.8
Solution

We may begin by letting \(x=u\text{,}\) \(0\leq u\leq 3\text{,}\) and \(y = 2-2u/3\text{.}\) This gives only the line on the “upper” side of the triangle. To get all of the region \(R\text{,}\) we can once again scale \(y\) by a variable factor, \(v\text{.}\)

Still letting \(x = u\text{,}\) \(0\leq u\leq 3\text{,}\) we let \(y = v(2-2u/3)\text{,}\) \(0\leq v\leq 1\text{.}\) When \(v=0\text{,}\) all \(y\)-values are 0, and we get the portion of the \(x\)-axis between \(x=0\) and \(x=3\text{.}\) When \(v=1\text{,}\) we get the upper side of the triangle. When \(v=1/2\text{,}\) we get the line \(y=1/2(2-2u/3) = 1-u/3\text{,}\) which is the line “halfway up” the triangle, shown in the figure with a dashed line.

Letting \(z = f(x,y) = x^2+2y^2\text{,}\) we have \(\vec r(u,v) = \langle u, v(2-2u/3), u^2+2\big(v(2-2u/3)\big)^2\rangle\text{,}\) \(0\leq u\leq 3\text{,}\) \(0\leq v\leq 1\text{.}\) This surface is graphed in Figure 16.5.9.(b). Again, when one looks from above, we can see the scaling effects of \(v\text{:}\) the series of lines that run to the point \((3,0)\) each represent a different value of \(v\text{.}\)

Another common way to parametrize the surface is to begin with \(y=u\text{,}\) \(0\leq u\leq 2\text{.}\) Solving the equation of the line \(y=2-2x/3\) for \(x\text{,}\) we have \(x = 3-3y/2\text{,}\) leading to using \(x=v(3-3u/2)\text{,}\) \(0\leq v\leq 1\text{.}\) With \(z=x^2+2y^2\text{,}\) we have \(\vec r(u,v) = \langle v(3-3u/2),u, \big(v(3-3u/2)\big)^2+2v^2\rangle\text{,}\) \(0\leq u\leq 2\text{,}\) \(0\leq v\leq 1\text{.}\)

Example 16.5.10. Parametrizing a surface over a triangle.

Parametrize the surface \(z=x^2+2y^2\) over the triangular region \(R\) enclosed by the lines \(y=3-2x/3\text{,}\) \(y=1\) and \(x=0\) as shown in Figure 16.5.11.(a).

(a)
(b)
Figure 16.5.11. Part (a) shows a graph of the region \(R\text{,}\) and part (b) shows the surface over \(R\text{,}\) as defined in Example 16.5.10
Solution

While the region \(R\) in this example is very similar to the region \(R\) in the previous example, and our method of parametrizing the surface is fundamentally the same, it will feel as though our answer is much different than before.

We begin with letting \(x=u\text{,}\) \(0\leq u\leq 3\text{.}\) We may be tempted to let \(y = v(3-2u/3)\text{,}\) \(0\leq v\leq 1\text{,}\) but this is incorrect. When \(v = 1\text{,}\) we obtain the upper line of the triangle as desired. However, when \(v=0\text{,}\) the \(y\)-value is 0, which does not lie in the region \(R\text{.}\)

We will describe the general method of proceeding following this example. For now, consider \(y = 1+v(2-2u/3)\text{,}\) \(0\leq v\leq 1\text{.}\) Note that when \(v=1\text{,}\) we have \(y=3-2u/3\text{,}\) the upper line of the boundary of \(R\text{.}\) Also, when \(v=0\text{,}\) we have \(y=1\text{,}\) which is the lower boundary of \(R\text{.}\) With \(z=x^2+2y^2\text{,}\) we determine \(\vec r(u,v) = \langle u, 1+v(2-2u/3), u^2+2\big(1+v(2-2u/3)\big)^2\rangle\text{,}\) \(0\leq u\leq 3\text{,}\) \(0\leq v\leq 1\text{.}\)

The surface is graphed in Figure 16.5.11.(b).

Given a surface of the form \(z=f(x,y)\text{,}\) one can often determine a parametrization of the surface over a region \(R\) in a manner similar to determining bounds of integration over a region \(R\text{.}\) Using the techniques of Section 15.1, suppose a region \(R\) can be described by \(a\leq x\leq b\text{,}\) \(g_1(x) \leq y\leq g_2(x)\text{,}\) i.e., the area of \(R\) can be found using the iterated integral

\begin{equation*} \int_a^b\int_{g_1(x)}^{g_2(x)}\, dy\, dx\text{.} \end{equation*}

When parametrizing the surface, we can let \(x=u\text{,}\) \(a\leq u\leq b\text{,}\) and we can let \(y = g_1(u)+v\big(g_2(u)-g_1(u)\big)\text{,}\) \(0\leq v\leq 1\text{.}\) The parametrization of \(x\) is straightforward, but look closely at how \(y\) is determined. When \(v=0\text{,}\) \(y=g_1(u) = g_1(x)\text{.}\) When \(v=1\text{,}\) \(y= g_2(u)=g_2(x)\text{.}\)

As a specific example, consider the triangular region \(R\) from Example 16.5.10, shown in Figure 16.5.11.(a). Using the techniques of Section 15.1, we can find the area of \(R\) as

\begin{equation*} \int_0^3\int_1^{3-2x/3} dy\, dx\text{.} \end{equation*}

Following the above discussion, we can set \(x=u\text{,}\) where \(0\leq u\leq 3\text{,}\) and set \(y = 1+ v\big(3-2u/3-1\big) = 1+v(2-2u/3)\text{,}\) \(0\leq v\leq 1\text{,}\) as used in that example.

One can do a similar thing if \(R\) is bounded by \(c\leq y\leq d\text{,}\) \(h_1(y)\leq x\leq h_2(y)\text{,}\) but for the sake of simplicity we leave it to the reader to flesh out those details. The principles outlined above are given in the following Key Idea for reference.

Key Idea 16.5.12. Parametrizing Surfaces.

Let a surface \(\surfaceS\) be the graph of a function \(f(x,y)\text{,}\) where the domain of \(f\) is a closed, bounded region \(R\) in the \(xy\)-plane. Let \(R\) be bounded by \(a\leq x\leq b\text{,}\) \(g_1(x)\leq y\leq g_2(x)\text{,}\) i.e., the area of \(R\) can be found using the iterated integral \(\int_a^b\int_{g_1(x)}^{g_2(x)}\, dy\, dx\text{,}\) and let \(h(u,v) = g_1(u)+v\big(g_2(u)-g_1(u)\big)\text{.}\)

\(\surfaceS\) can be parametrized as

\begin{equation*} \vec r(u,v) = \la u, h(u,v), f\big(u,h(u,v)\big)\ra, a\leq u\leq b,\ 0\leq v\leq 1\text{.} \end{equation*}
Example 16.5.13. Parametrizing a cylindrical surface.

Find a parametrization of the cylinder \(x^2 + z^2/4=1\text{,}\) where \(-1\leq y\leq 2\text{,}\) as shown in Figure 16.5.14.

Figure 16.5.14. The cylinder parametrized in Example 16.5.13
Solution

The equation \(x^2+z^2/4=1\) can be envisioned to describe an ellipse in the \(xz\)-plane; as the equation lacks a \(y\)-term, the equation describes a cylinder (recall Definition 11.1.13) that extends without bound parallel to the \(y\)-axis. This ellipse has a vertical major axis of length 4, a horizontal minor axis of length 2, and is centered at the origin. We can parametrize this ellipse using sines and cosines; our parametrization can begin with

\begin{equation*} \vec r(u,v) = \la \cos u, \text{ ??? } , 2\sin u\ra, 0\leq u\leq 2\pi\text{,} \end{equation*}

where we still need to determine the \(y\) component.

While the cylinder \(x^2+z^2/4=1\) is satisfied by any \(y\) value, the problem states that all \(y\) values are to be between \(y=-1\) and \(y=2\text{.}\) Since the value of \(y\) does not depend at all on the values of \(x\) or \(z\text{,}\) we can use another variable, \(v\text{,}\) to describe \(y\text{.}\) Our final answer is

\begin{equation*} \vec r(u,v) = \la \cos u, v, 2\sin u\ra, 0\leq u\leq 2\pi, -1\leq v\leq 2\text{.} \end{equation*}
Example 16.5.15. Parametrizing an elliptic cone.

Find a parametrization of the elliptic cone \(z^2 = \frac{x^2}{4}+\frac{y^2}{9}\text{,}\) where \(-2\leq z\leq 3\text{,}\) as shown in Figure 16.5.16.

Figure 16.5.16. The elliptic cone as described in Example 16.5.15
Solution

One way to parametrize this cone is to recognize that given a \(z\) value, the cross section of the cone at that \(z\) value is an ellipse with equation \(\frac{x^2}{(2z)^2} + \frac{y^2}{(3z)^2}=1\text{.}\) We can let \(z=v\text{,}\) for \(-2\leq v\leq 3\) and then parametrize the above ellipses using sines, cosines and \(v\text{.}\)

We can parametrize the \(x\) component of our surface with \(x=2z\cos u\) and the \(y\) component with \(y=3z\sin u\text{,}\) where \(0\leq u\leq 2\pi\text{.}\) Putting all components together, we have

\begin{equation*} \vec r(u,v) = \la 2v\cos u, 3v\sin u, v\ra, 0\leq u\leq 2\pi, -2\leq v\leq 3\text{.} \end{equation*}

When \(v\) takes on negative values, the radii of the cross-sectional ellipses become “negative,” which can lead to some surprising results. Consider Figure 16.5.17, where the cone is graphed for \(0\leq u\leq \pi\text{.}\) Because \(v\) is negative below the \(xy\)-plane, the radii of the cross-sectional ellipses are negative, and the opposite side of the cone is sketched below the \(xy\)-plane.

Figure 16.5.17. The elliptic cone as described in Example 16.5.15 with restricted domain
Example 16.5.18. Parametrizing an ellipsoid.

Find a parametrization of the ellipsoid \(\frac{x^2}{25}+y^2+\frac{z^2}{4}=1\) as shown in Figure 16.5.19.(a).

(a)
(b)
Figure 16.5.19. An ellipsoid in (a), drawn again in (b) with its domain restricted, as described in Example 16.5.18
Solution

Recall Key Idea 11.2.20 from Section 11.2, which states that all unit vectors in space have the form \(\langle \sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta\rangle\) for some angles \(\theta\) and \(\varphi\text{.}\) If we choose our angles appropriately, this allows us to draw the unit sphere. To get an ellipsoid, we need only scale each component of the sphere appropriately.

The \(x\)-radius of the given ellipsoid is 5, the \(y\)-radius is 1 and the \(z\)-radius is 2. Substituting \(u\) for \(\theta\) and \(v\) for \(\varphi\text{,}\) we have

\begin{equation*} \vec r(u,v) = \langle 5\sin u\cos v, \sin u\sin v,2\cos u\rangle\text{,} \end{equation*}

where we still need to determine the ranges of \(u\) and \(v\text{.}\)

Note how the \(x\) and \(y\) components of \(\vec r\) have \(\cos v\) and \(\sin v\) terms, respectively. This hints at the fact that ellipses are drawn parallel to the \(xy\)-plane as \(v\) varies, which implies we should have \(v\) range from \(0\) to \(2\pi\text{.}\)

One may be tempted to let \(0\leq u\leq 2\pi\) as well, but note how the \(z\) component is \(2\cos u\text{.}\) We only need \(\cos u\) to take on values between \(-1\) and \(1\) once, therefore we can restrict \(u\) to \(0\leq u\leq \pi\text{.}\)

The final parametrization is thus

\begin{equation*} \vec r(u,v) = \langle 5\sin u\cos v, \sin u\sin v,2\cos u\rangle, 0\leq u\leq\pi, 0\leq v\leq 2\pi\text{.} \end{equation*}

In Figure 16.5.19.(b), the ellipsoid is graphed on \(\frac{\pi}{4}\leq u\leq \frac{2\pi}{3}\text{,}\) \(\frac{\pi}4\leq v\leq \frac{3\pi}2\) to demonstrate how each variable affects the surface.

Parametrization is a powerful way to represent surfaces. One of the advantages of the methods of parametrization described in this section is that the domain of \(\vec r(u,v)\) is always a rectangle; that is, the bounds on \(u\) and \(v\) are constants. This will make some of our future computations easier to evaluate.

Just as we could parametrize curves in more than one way, there will always be multiple ways to parametrize a surface. Some ways will be more “natural” than others, but these other ways are not incorrect. Because technology is often readily available, it is often a good idea to check one's work by graphing a parametrization of a surface to check if it indeed represents what it was intended to.

Subsection 16.5.2 Surface Area

Figure 16.5.20. Tangent and normal vectors for parametric surfaces

It will become important in the following sections to be able to compute the surface area of a surface \(\surfaceS\) given a smooth parametrization \(\vec r(u,v)\text{,}\) \(a\leq u\leq b\text{,}\) \(c\leq v\leq d\text{.}\) Following the principles given in the integration review at the beginning of this chapter, we can say that

\begin{equation*} \text{ Surface Area of }\surfaceS\, =S = \iint_{\surfaceS}\, dS\text{,} \end{equation*}

where \(dS\) represents a small amount of surface area. That is, to compute total surface area \(S\text{,}\) add up lots of small amounts of surface area \(dS\) across the entire surface \(\surfaceS\text{.}\) The key to finding surface area is knowing how to compute \(dS\text{.}\) We begin by approximating.

In Section 15.5 we used the area of a plane to approximate the surface area of a small portion of a surface. We will do the same here.

Let \(R\) be the region of the \(u\)-\(v\) plane bounded by \(a\leq u\leq b\text{,}\) \(c\leq v\leq d\) as shown in Figure 16.5.21.(a). Partition \(R\) into rectangles of width \(\Delta u = \frac{b-a}n\) and height \(\Delta v = \frac{d-c}n\text{,}\) for some \(n\text{.}\) Let \(p=(u_0,v_0)\) be the lower left corner of some rectangle in the partition, and let \(m\) and \(q\) be neighboring corners as shown.

The point \(p\) maps to a point \(P = \vec r(u_0,v_0)\) on the surface \(\surfaceS\text{,}\) and the rectangle with corners \(p\text{,}\) \(m\) and \(q\) maps to some region (probably not rectangular) on the surface as shown in Figure 16.5.21.(b), where \(M = \vec r(m)\) and \(Q = \vec r(q)\text{.}\) We wish to approximate the surface area of this mapped region.

Let \(\vec u = M-P\) and \(\vec v = Q-P\text{.}\) These two vectors form a parallelogram, illustrated in Figure 16.5.21.(c), whose area approximates the surface area we seek. In this particular illustration, we can see that parallelogram does not particularly match well the region we wish to approximate, but that is acceptable; by increasing the number of partitions of \(R\text{,}\) \(\Delta u\) and \(\Delta v\) shrink and our approximations will become better.

(a)
(b)
(c)
Figure 16.5.21. Illustrating the process of finding surface area by approximating with planes

From Section 11.4 we know the area of this parallelogram is \(\snorm{\vec u\times \vec v}\text{.}\) If we repeat this approximation process for each rectangle in the partition of \(R\text{,}\) we can sum the areas of all the parallelograms to get an approximation of the surface area \(S\text{:}\)

\begin{equation*} \text{ Surface area of } \surfaceS \, =S \approx \sum_{j=1}^n\sum_{i=1}^n \snorm{\vec u_{i,j}\times \vec v_{i,j}}\text{,} \end{equation*}

where \(\vec u_{i,j} = \vec r(u_i+\Delta u,v_j) - \vec r(u_i,v_j)\) and \(\vec v_{i,j} = \vec r(u_i,v_j+\Delta v)-\vec r(u_i,v_j)\text{.}\)

From our previous calculus experience, we expect that taking a limit as \(n\to \infty\) will result in the exact surface area. However, the current form of the above double sum makes it difficult to realize what the result of that limit is. The following rewriting of the double summation will be helpful:

\begin{align*} \amp \sum_{j=1}^n\sum_{i=1}^n \snorm{\vec u_{i,j}\times \vec v_{i,j}}=\\ \amp \sum_{j=1}^n\sum_{i=1}^n \snorm{\big(\vec r(u_i+\Delta u,v_j) - \vec r(u_i,v_j)\big) \times \big(\vec r(u_i,v_j+\Delta v)-\vec r(u_i,v_j)\big)}=\\ \amp \sum_{j=1}^n\sum_{i=1}^n \snorm{\frac{\vec r(u_i+\Delta u,v_j) - \vec r(u_i,v_j)}{\Delta u} \times \frac{\vec r(u_i,v_j+\Delta v)-\vec r(u_i,v_j)}{\Delta v}}\Delta u\Delta v\text{.} \end{align*}

We now take the limit as \(n\to\infty\text{,}\) forcing \(\Delta u\) and \(\Delta v\) to 0. As \(\Delta u\to 0\text{,}\)

\begin{equation*} \frac{\vec r(u_i+\Delta u,v_j) - \vec r(u_i,v_j)}{\Delta u} \to \vec r_u(u_i,v_j) \text{ and } \end{equation*}
\begin{equation*} \frac{\vec r(u_i,v_j+\Delta v)-\vec r(u_i,v_j)}{\Delta v} \to \vec r_v(u_i,v_j)\text{.} \end{equation*}

(This limit process also demonstrates that \(\vec r_u(u,v)\) and \(\vec r_v(u,v)\) are tangent to the surface \(\surfaceS\) at \(\vec r(u,v)\text{.}\) We don't need this fact now, but it will be important in the next section.)

Thus, in the limit, the double sum leads to a double integral:

\begin{equation*} \lim_{n\to\infty} \sum_{j=1}^n\sum_{i=1}^n \snorm{\vec u_{i,j}\times \vec v_{i,j}}= \int_c^d\int_a^b \snorm{\vec r_u\times\vec r_v}\, du\, dv\text{.} \end{equation*}
Figure 16.5.23. Area of parametric surfaces
Example 16.5.24. Finding the surface area of a parametrized surface.

Using the parametrization found in Example 16.5.6, find the surface area of \(z=x^2+2y^2\) over the circular disk of radius 2, centered at the origin.

Solution

In Example 16.5.6, we parametrized the surface as \(\vec r(u,v) = \la 2v\cos u, 2v\sin u, 4v^2\cos^2u+8v^2\sin^2u\ra\text{,}\) for \(0\leq u\leq 2\pi\text{,}\) \(0\leq v\leq 1\text{.}\) To find the surface area using Theorem 16.5.22, we need \(\snorm{\vec r_u\times\vec r_v}\text{.}\) We find:

\begin{align*} \vec r_u \amp = \la -2v\sin u, 2v\cos u, 8v^2\cos u\sin u\ra\\ \vec r_v \amp = \la 2\cos u, 2\sin v, 8v\cos^2 u+16v\sin^2u\ra\\ \vec r_u\times\vec r_v \amp = \la 16v^2\cos u, 32v^2\sin u, -4v\ra\\ \snorm{\vec r_u\times\vec r_v} \amp = \sqrt{256v^4\cos^2u+1024v^4\sin^2u+16v^2}\text{.} \end{align*}

Thus the surface area is

\begin{align*} S = \iint_\surfaceS \, dS \amp = \iint_R\snorm{\vec r_u\times \vec r_v}\, dA\\ \amp = \int_0^1\int_0^{2\pi} \sqrt{256v^4\cos^2u+1024v^4\sin^2u+16v^2}\, du\, dv\\ \amp \approx 53.59\text{.} \end{align*}

There is a lot of tedious work in the above calculations and the final integral is nontrivial. The use of a computer-algebra system is highly recommended.

Figure 16.5.25. Surface area of a sphere

In Section 16.1, we recalled the arc length differential \(ds=\snorm{\vrp(t)}dt\text{.}\) In subsequent sections, we used that differential, but in most applications the “\(\snorm{\vrp(t)}\)” part of the differential canceled out of the integrand (to our benefit, as integrating the square roots of functions is generally difficult). We will find a similar thing happens when we use the surface area differential \(dS\) in the following sections. That is, our main goal is not to be able to compute surface area; rather, surface area is a tool to obtain other quantities that are more important and useful. In our applications, we will use \(dS\text{,}\) but most of the time the “\(\snorm{\vec r_u\times \vec r_v}\)” part will cancel out of the integrand, making the subsequent integration easier to compute.

Exercises 16.5.3 Exercises

Terms and Concepts
1.

In your own words, describe what an orientable surface is.

2.

Give an example of a non-orientable surface.

Problems

In the following exercises, parametrize the surface defined by the function \(z=f(x,y)\) over each of the given regions \(R\) of the \(xy\)-plane.

3.

\(z = 3x^2y\text{;}\)

  1. \(R\) is the rectangle bounded by \(-1\leq x\leq 1\) and \(0\leq y\leq 2\text{.}\)

  2. \(R\) is the circle of radius 3, centered at \((1,2)\text{.}\)

  3. \(R\) is the triangle with vertices \((0,0)\text{,}\) \((1,0)\) and \((0,2)\text{.}\)

  4. \(R\) is the region bounded by the \(x\)-axis and the graph of \(y = 1-x^2\text{.}\)

4.

\(z = 4x+2y^2\text{;}\)

  1. \(R\) is the rectangle bounded by \(1\leq x\leq 4\) and \(5\leq y\leq 7\text{.}\)

  2. \(R\) is the ellipse with major axis of length 8 parallel to the \(x\)-axis, and minor axis of length 6 parallel to the \(y\)-axis, centered at the origin.

  3. \(R\) is the triangle with vertices \((0,0)\text{,}\) \((2,2)\) and \((0,4)\text{.}\)

  4. \(R\) is the annulus bounded between the circles, centered at the origin, with radius 2 and radius 5.

In the following exercises, a surface \(\surfaceS\) in space is described that cannot be defined as the graph of a function \(f(x,y)\text{.}\) Give a parametrization of \(\surfaceS\text{.}\)

5.

\(\surfaceS\) is the rectangle in space with corners at \((0,0,0)\text{,}\) \((0,2,0)\text{,}\) \((0,2,1)\) and \((0,0,1)\text{.}\)

6.

\(\surfaceS\) is the triangle in space with corners at \((1,0,0)\text{,}\) \((1,0,1)\) and \((0,0,1)\text{.}\)

7.

\(\surfaceS\) is the ellipsoid \(\ds\frac{x^2}{9} + \frac{y^2}{4}+\frac{z^2}{16} = 1\text{.}\)

8.

\(\surfaceS\) is the elliptic cone \(\ds y^2= x^2+\frac{z^2}{16}\text{,}\) for \(-1 \leq y \leq 5\text{.}\)

In the following exercises, a domain \(D\) in space is given. Parametrize each of the bounding surfaces of \(D\text{.}\)

9.

\(D\) is the domain bounded by the planes \(z = \frac12(3-x)\text{,}\) \(x=1\text{,}\) \(y=0\text{,}\) \(y=2\) and \(z=0\text{.}\)

10.

\(D\) is the domain bounded by the planes \(z=2x+4y-4\text{,}\) \(x=2\text{,}\) \(y=1\) and \(z=0\text{.}\)

11.

\(D\) is the domain bounded by \(z=2y\text{,}\) \(y=4-x^2\) and \(z=0\text{.}\)

12.

\(D\) is the domain bounded by \(y=1-z^2\text{,}\) \(y=1-x^2\text{,}\) \(x=0\text{,}\) \(y=0\) and \(z=0\text{.}\)

13.

\(D\) is the domain bounded by the cylinder \(x+y^2/9=1\) and the planes \(z=1\) and \(z=3\text{.}\)

14.

\(D\) is the domain bounded by the cone \(x^2+y^2=(z-1)^2\) and the plane \(z=0\text{.}\)

15.

\(D\) is the domain bounded by the cylinder \(z=1-x^2\) and the planes \(y=-1\text{,}\) \(y=2\) and \(z=0\text{.}\)

16.

\(D\) is the domain bounded by the paraboloid \(z=4-x^2-4y^2\) and the plane \(z=0\text{.}\)

In the following exercises, find the surface area \(S\) of the given surface \(\surfaceS\text{.}\) (The associated integrals are computable without the assistance of technology.)

17.

\(\surfaceS\) is the plane \(z=2x+3y\) over the rectangle \(-1\leq x\leq 1\text{,}\) \(2\leq v \leq 3\text{.}\)

18.

\(\surfaceS\) is the plane \(z=x+2y\) over the triangle with vertices at \((0,0)\text{,}\) \((1,0)\) and \((0,1)\text{.}\)

19.

\(\surfaceS\) is the plane \(z=x+y\) over the circular disk, centered at the origin, with radius 2.

20.

\(\surfaceS\) is the plane \(z=x+y\) over the annulus bounded by the circles, centered at the origin, with radius 1 and radius 2.

In the following exercises, set up the double integral that finds the surface area \(S\) of the given surface \(\surfaceS\text{,}\) then use technology to approximate its value.

21.

\(\surfaceS\) is the paraboloid \(z=x^2+y^2\) over the circular disk of radius 3 centered at the origin.

22.

\(\surfaceS\) is the paraboloid \(z=x^2+y^2\) over the triangle with vertices at \((0,0)\text{,}\) \((0,1)\) and \((1,1)\text{.}\)

23.

\(\surfaceS\) is the plane \(z=5x-y\) over the region enclosed by the parabola \(y=1-x^2\) and the \(x\)-axis.

24.

\(\surfaceS\) is the hyperbolic paraboloid \(z=x^2-y^2\) over the circular disk of radius 1 centered at the origin.