1.6 Integrals of Scalar Functions Over Surfaces

393

Now we are ready to define the integral of a scalar function \(f\) over a surface \(S\). This concept is a natural generalization of the area of a surface, which corresponds to the integral over \(S\) of the scalar function \(f(x, y, z) = 1\). This is quite analogous to considering the path integral as a generalization of arc length. In the next section we shall deal with the integral of a vector function \({\bf F}\) over a surface. These concepts will play a crucial role in the vector analysis treated in the final chapter.

Let us start with a surface \(S\) parametrized by a mapping \({\Phi}\colon\, D\to S \subset {\mathbb R}^3\), where D is an elementary region, which we write as \({\Phi}(u,v) = (x(u,v),y(u,v), z(u,v))\).

Definition: The Integral of a Scalar Function Over a Surface

If \(f(x, y, z)\) is a real-valued continuous function defined on a parametrized surface \(S\), we define the integral of f over \(S\) to be \begin{equation*} \int\!\!\!\int\nolimits_{S}f(x,y,z) \,dS = \int\!\!\!\int\nolimits_{S} f \,dS = \int\!\!\!\int\nolimits_{D} f({\Phi}(u,v)) \|{\bf T}_u \times {\bf T}_v\| \,du \,dv.\tag{1} \end{equation*}

Written out, equation (1) becomes \begin{equation*} \begin{array}{lll} \int\!\!\!\int\nolimits_{S}f\,dS &=& \int\!\!\!\int\nolimits_{D}f(x(u,v),y(u,v),z(u,v)) \,\sqrt{\bigg[\frac{\partial(x,y)}{\partial(u,v)}\bigg]^2 +\bigg[\frac{\partial(y,z)}{\partial(u,v)}\bigg]^2 + \bigg[\frac{\partial(x,z)}{\partial(u,v)}\bigg]^2} d u \,dv. \end{array}\tag{2} \end{equation*}

Thus, if \(f\) is identically 1, we recover the area formula (3) of Section 1.5. Like surface area, the surface integral is independent of the particular parametrization used. This will be discussed in Section 1.7.

Figure 1.38: \({\Phi}\) takes a portion \(R_{\it ij}\) of \(D\) to a portion of \(S\).

We can gain some intuitive knowledge about this integral by considering it as a limit of sums. Let \(D\) be a rectangle partitioned into \(n^2\) rectangles \(R_{\it ij}\) with areas \(\Delta u\, \Delta v\). Let \(S_{\it ij}={\Phi}(R_{\it ij})\) be the portion of the surface \({\Phi}(D)\) corresponding to \(R_{\it ij}\) (see Figure 1.38), and let \(A(S_{\it ij})\) be the area of this portion of the surface. For large \(n, f\) will be approximately constant on \(S_{\it ij}\), and we form the sum \begin{equation*} S_n = \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} f({\Phi}(u_i,v_j))A(S_{\it ij}),\tag{3} \end{equation*} where \((u_i,v_j)\in R_{\it ij}\). From Section 1.5 we have a formula for \(A(S_{\it ij}) \): \[ A(S_{\it ij}) = \int\!\!\!\int\nolimits_{R_{\it ij}} \|{\bf T}_u \times {\bf T}_v\| \,du \,dv, \] which, by the mean-value theorem for integrals, equals \(\|{\bf T}_{u_i^*} \times {\bf T}_{v_j^*}\|\, { \Delta} u \,{ \Delta} v\) for some point \((u_i^*,v_j^*)\) in \(R_{\it ij}\). Hence, our sum becomes \[ S_n = \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} f({\Phi}(u_i,v_j)) \|{\bf T}_{u_i^*} \times {\bf T}_{v_j^*}\| { \Delta} u\, { \Delta} v, \] which is an approximating sum for the last integral in formula (1). Therefore, \[ {\mathop {\rm limit}_{n\to \infty}\ } S_n = \int\!\!\!\int\nolimits_{S} f \,dS. \]

394

Note that each term in the sum in formula (3) is the value of \(f\) at some point \({\Phi}(u_i,v_j)\) times the area of \(S_{\it ij}\). Compare this with the Riemann-sum interpretation of the path integral in Section 1.2.

If \(S\) is a union of parametrized surfaces \(S_i,\ i=1,\ldots,N\), that do not intersect except possibly along curves defining their boundaries, then the integral of \(f\) over \(S\) is defined by \[ \int\!\!\!\int\nolimits_{S} f\,dS = \sum_{i=1}^N \int\!\!\!\int\nolimits_{S_i} f \,dS, \] as we should expect. For example, the integral over the surface of a cube may be expressed as the sum of the integrals over the six sides.

example 1

Suppose a helicoid is described as in Example 2, Section 1.5, and let \(f\) be given by \(f(x,y,z) = \sqrt{x^2+y^2 +1}\). Find \({\int\!\!\!\int}_S f \,dS\).

solution As in Examples 1 and 2 of Section 1.5, \[ \frac{\partial(x,y)}{\partial(r,\theta)} = r, \qquad \frac{\partial(y,z)}{\partial(r,\theta)} =\sin \theta,\qquad \frac{\partial(x,z)}{\partial(r,\theta)} = \cos\theta. \]

Also, \(f(r\cos\theta,r\sin\theta,\theta) = \sqrt{r^2 + 1}\). Therefore, \begin{eqnarray*} \int\!\!\!\int\nolimits_{S}f(x,y,z) \,dS &=& \int\!\!\!\int\nolimits_{D} f({\Phi}(r,\theta))\|{\bf T}_r \times {\bf T}_{\theta}\| \,dr \,d\theta\\[5pt] &=& \int_0^{2\pi} \int_0^1 \sqrt{r^2 + 1} \sqrt{r^2 + 1} \,dr \,d\theta = \int_0^{2\pi} \frac{4}{3} \,d\theta = \frac{8}{3} \pi.\\[-28pt] \end{eqnarray*}

Question 1.114 Section 17.5 Progress Check Question #1

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
3
Correct.
Try again. Note that this surface was discussed in Example 1 in Section 1.5.
Incorrect.

Surface Integrals Over Graphs

Suppose \(S\) is the graph of a \(C^1\) function \(z=g(x,y)\). Recall from Section 1.5 that we can parametrize \(S\) by \[ x=u,\qquad y = v,\qquad z = g(u,v), \] and that in this case \[ \|{\bf T}_u \times {\bf T}_v\| = \sqrt{1+\bigg(\frac{\partial g}{\partial u}\bigg)^2 + \bigg(\frac{\partial g}{\partial v}\bigg)^2}, \] so \begin{equation*} \int\!\!\!\int\nolimits_{S} f(x,y,z) \,dS = \int\!\!\!\int\nolimits_{D} f(x,y,g(x,y)) \sqrt{1+\bigg(\frac{\partial g}{\partial x}\bigg)^2+ \bigg(\frac{\partial g}{\partial y}\bigg)^2} \,{\it dx} \,{\it dy}.\tag{4} \end{equation*}

395

example 2

Let \(S\) be the surface defined by \(z = x^2 + y\), where \(D\) is the region \(0 \le x \le 1, -1 \le y \le 1\). Evaluate \({\int\!\!\!\int}_S x \,dS\).

solution If we let \(z = g(x,y) = x^2 + y\), formula (4) gives \begin{eqnarray*} \int\!\!\!\int\nolimits_{S} x \,dS &=& \int\!\!\!\int\nolimits_{D} x \sqrt{1+\bigg(\frac{\partial g}{\partial x}\bigg)^2 + \bigg(\frac{\partial g}{\partial y}\bigg)^2} \,{\it dx} \,{\it dy} = \int_{-1}^1 \int_0^1 x \sqrt{1 + 4x^2 + 1} \,{\it dx} \,{\it dy} \\[5pt] &=& \frac{1}{8} \int_{-1}^1 \bigg[\int_0^1 (2+4x^2)^{1/2} (8x\,{\it dx} )\bigg] {\it dy} = \frac23 \,\, {\cdot}\, \, \frac18 \int_{-1}^1 [(2 + 4x^2)^{3/2}]|^1_0 \,{\it dy} \\[5pt] & = & \frac{1}{12} \int_{-1}^1 (6^{3/2} - 2^{3/2}) \,{\it dy} = \frac{1}{6}(6^{3/2}- 2^{3/2}) = \sqrt{6} - \frac{\sqrt{2}}{3}\\ [5pt] & = & \sqrt{2} \bigg(\sqrt{3} - \frac{1}{3}\bigg).\\[-28pt] \end{eqnarray*}

Question 1.115 Section 17.5 Progress Check Question #2

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3
Correct.
Try again. You can use the fact that for this graph given by \(z=g(x,y), dS = \sqrt{1+\bigg(\frac{\partial g}{\partial x}\bigg)^2+ \bigg(\frac{\partial g}{\partial y}\bigg)^2} \,{\it dx} \,{\it dy}\).
Incorrect.

example 3

Evaluate \(\int\!\!\!\int\nolimits_{\! S} z^2 \,dS\), where \(S\) is the unit sphere \(x^2 + y^2 + z^2 =1\).

solution For this problem, it is convenient to use spherical coordinates and to represent the sphere parametrically by the equations \(x = \cos\theta\sin\phi,y=\sin\theta\sin\phi,z=\cos\phi\), over the region \(D\) in the \(\theta\phi\) plane given by the inequalities \(0\le \phi \le \pi, 0\le \theta \le 2\pi\). From equation (1) we get \[ \int\!\!\!\int\nolimits_{S} z^2 \,dS = \int\!\!\!\int\nolimits_{D} (\cos \phi)^2 \|{\bf T}_{\theta}\times {\bf T}_{\phi}\| \,d\theta \,d\phi. \]

A little computation [use formula (2) of Section 1.5; see Exercise 12] shows that \[ \|{\bf T}_{\theta}\times {\bf T}_{\phi}\| = \sin \phi . \] (Note that for \(0\leq\phi\leq \pi, \hbox{ we have } \sin \phi\geq 0.)\) Thus, \begin{eqnarray*} \int\!\!\!\int\nolimits_{S} z^2 \,dS &=& \int_0^{2\pi} \int_0^{\pi} \cos^2 \phi \sin\phi \,d\phi \,d\theta \\[4pt] & = & \frac{1}{3} \int_0^{2\pi} [-{\rm cos}^3 \phi]_0^{\pi} \,d\theta = \frac{2}{3} \int_0^{2\pi} \,d\theta = \frac{4\pi}{3}.\\[-28pt] \end{eqnarray*}

This example also shows that on a sphere of radius R, \[ \int\!\!\!\int_{S} f \ {\it ds} = \int_{0}^{2\pi} \int_{0}^{\pi} f(\phi,\theta) {\it R}^2 \sin \phi \, d\phi \, d\theta, \] or, for short, the area element on the sphere is given by \[ dS = {\it R}^2 \sin\phi \, d\phi \, d\theta. \]

Question 1.116 Section 17.3 Progress Check Question #3

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3
Correct.
Try again. The area element \(dS\) is equal to \(\sin (\phi) d \phi d \theta\).
Incorrect.

Integrals Over Graphs

396

We now develop another formula for surface integrals when the surface can be represented as a graph. To do so, we let \(S\) be the graph of \(z =g(x, y)\) and consider formula (4). We claim that \begin{equation*} \int\!\!\!\int\nolimits_{S} f(x,y,z) \,dS = \int\!\!\!\int\nolimits_{D}\frac{f(x,y,g(x,y))}{\cos\theta}\,{\it dx} \,{\it dy},\tag{5} \end{equation*} where \(\theta\) is the angle the normal to the surface makes with the unit vector \({\bf k}\) at the point \((x, y, g(x, y))\) (see Figure 1.39). Describing the surface by the equation \(\phi(x,y,z) = z-g(x,y) = 0\), a normal vector \({\bf N}\) is \(\nabla\phi\); that is, \begin{equation*} {\bf N} = -\frac{\partial g}{\partial x}{\bf i} - \frac{\partial g}{\partial y}{\bf j} + {\bf k}\tag{6} \end{equation*}

Figure 1.39: The area of a patch of area \({ \Delta} S\) over a patch \({ \Delta} A\) is \({ \Delta} S = { \Delta} A/\cos \theta\), where \(\theta\) is the angle the unit normal \({\bf n}\) makes with \({\bf k}\).

[see Example 4 of Section 1.4, or recall that the normal to a surface with equation \(g(x, y, z) = \hbox{constant is given by } \nabla g\)]. Thus, \[ \cos\theta = \frac{{\bf N}\, {\cdot}\, {\bf k}}{\|{\bf N}\|} = \frac{1}{\sqrt{(\partial g/\partial x)^2+(\partial g/\partial y)^2 +1}} . \]

Substitution of this formula into equation (4) gives equation (5). Note that \({\cos\theta={\bf n} \,{\cdot}\, {\bf k}}\), where \({{\bf n}={\bf N}/\|{\bf N}\|}\) is the unit normal. Thus, we can write \[ d{\bf S}= \frac{{\it dx}\ {\it dy}}{{\bf n}\, {\cdot}\,{\bf k}}. \]

The result is, in fact, obvious geometrically, for if a small rectangle in the \({\it xy}\) plane has area \({\Delta} A\), then the area of the portion above it on the surface is \({\Delta} S = { \Delta} A{/}\cos\theta\) (Figure 1.39). This intuitive approach can help us to remember formula (5) and to apply it in problems.

397

example 4

Compute \({\int\!\!\!\int}_S x \,dS\), where \(S\) is the triangle with vertices \((1,0,0), (0, 1,0), (0, 0, 1)\) (see Figure 1.40).

Figure 1.40: In computing a specific surface integral, we find a formula for the unit normal \({\bf n}\) and computes the angle \(\theta\) in preparation for formula (5).

solution This surface is the plane described by the equation \(x + y + z = 1\). Because the surface is a plane, the angle \(\theta\) is constant and a unit normal vector is \({\bf n} = (1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})\). Thus, \(\cos\theta ={\bf n}{\, {\cdot}\, }{\bf k} = 1/\sqrt{3}\), and by equation (5), \[ \int\!\!\!\int\nolimits_{S} x \,dS = \sqrt{3}\int\!\!\!\int\nolimits_{D} x \,{\it dx} \,{\it dy} , \] where \(D\) is the domain in the \({\it xy}\) plane. But \[ \sqrt{3}\int\!\!\!\int\nolimits_{D} x\,{\it dx} \,{\it dy} = \sqrt{3} \int_0^1 \int_0^{1-x} x \,{\it dy}\,{\it dx} = \sqrt{3}\int_0^1 x(1-x)\,{\it dx} = \frac{\sqrt{3}}{6}. \]

Integrals of functions over surfaces are useful for computing the mass of a surface when the mass density function \(m\) is known. The total mass of a surface with mass density (per unit area) \(m\) is given by \begin{equation*} M(S\,) = \int\!\!\!\int\nolimits_{S} m(x,y,z)\,dS.\tag{7} \end{equation*}

example 5

Let \({\Phi}\colon\, D \to {\mathbb R}^3\) be the parametrization of the helicoid \(S = {\Phi}(D)\) of Example 2 of Section 1.5. Recall that \({\Phi}(r,\theta) = (r\cos\theta, r \sin\theta,\theta)\), where \(0 \le \theta \le 2\pi\), and \(0 \le r \le 1\). Suppose \(S\) has a mass density at \((x, y, z)\in S\) equal to twice the distance of \((x, y, z)\) from the central axis (see Figure 1.31), that is, \(m(x, y, z)=2\sqrt{x^2 + y^2} = 2r\), in the cylindrical coordinate system. Find the total mass of the surface.

solution Applying formula (7), \[ M(S\,) = \int\!\!\!\int\nolimits_{S} 2 \sqrt{x^2 + y^2}\,d S = \int\!\!\!\int\nolimits_{D} 2r \,dS = \int\!\!\!\int\nolimits_{D} 2r\|{\bf T}_r \times {\bf T}_{\theta}\| \,dr \,d\theta. \]

From Example 2 of Section 1.5, we see that \(\|{\bf T}_r \times {\bf T}_{\theta}\| = \sqrt{1 + r^2}\). Thus, \begin{eqnarray*} M(S\,) &=& \int\!\!\!\int\nolimits_{D} 2r \sqrt{1 + r^2} \,dr \,d\theta = \int_0^{2\pi}\int_0^1 2r\sqrt{1+r^2} \,dr \,d\theta\\[5pt] &=& \int_0^{2\pi} \left[ \frac23 (1+r^2)^{3/2}\right]_0^1 \,d\theta = \int_0^{2\pi} \frac23 \, (2^{3/2} - 1) \,d\theta = \frac{4\pi}{3} (2^{3/2} - 1).\\[-28pt] \end{eqnarray*}