APEX Calculus

Consider the surface and curve shown in Figure 15.1.2.(a). The surface is given by $f(x,y)=1-\cos (x)\sin (y)\text{.}$ The dashed curve lies in the $xy$-plane and is the familiar $y=x^2$ parabola from $-1\le x\le 1\text{;}$ we’ll call this curve $C\text{.}$ The curve drawn with a solid line in the graph is the curve in space that lies on our surface with $x$ and $y$ values that lie on $C\text{.}$

The question we want to answer is this: what is the area that lies below the curve drawn with the solid line? In other words, what is the area of the region above $C$ and under the the surface $z=f(x,y)\text{?}$ This region is shown in Figure 15.1.2.(b).

We suspect the answer can be found using an integral, but before trying to figure out what that integral is, let us first try to approximate its value.

In Figure 15.1.2.(c), four rectangles have been drawn over the curve $C\text{.}$ The bottom corners of each rectangle lie on $C\text{,}$ and each rectangle has a height given by the function $f(x,y)$ for some $(x,y)$ pair along $C$ between the rectangle’s bottom corners.

As we know how to find the area of each rectangle, we are able to approximate the area above $C$ and under $f\text{.}$ Clearly, our approximation will be an approximation. The heights of the rectangles do not match exactly with the surface $f\text{,}$ nor does the base of each rectangle follow perfectly the path of $C\text{.}$

When first learning of the integral, and approximating areas with “heights × widths”, the width was a small change in $x\text{:}$ $dx\text{.}$ That will not suffice in this context. Rather, each width of a rectangle is actually approximating the arc length of a small portion of $C\text{.}$ In Section 12.5, we used $s$ to represent the arc-length parameter of a curve. A small amount of arc length will thus be represented by $ds\text{.}$

Here we have introduce a new notation, the integral symbol with a subscript of $C\text{.}$ It is reminiscent of our usage of ${\iint }_R\text{.}$ Using the train of thought found in the Integration Review preceding this section, we interpret “${\int }_{C}f(s)ds$” as meaning “sum up, along a curve $C\text{,}$ function values $f(s)\times$small arc lengths.” It is understood here that $s$ represents the arc-length parameter.

All this leads us to a definition. The integral found in Equation (15.1.1) is called a line integral. We formally define it below, but note that the definition is very abstract. On one hand, one is apt to say “the definition makes sense,” while on the other, one is equally apt to say “but I don’t know what I’m supposed to do with this definition.” We’ll address that after the definition, and actually find an answer to the area problem we posed at the beginning of this section.

The definition of the line integral does not specify whether $C$ is a curve in the plane or space (or hyperspace), as the definition holds regardless. For now, we’ll assume $C$ lies in the $xy$-plane.

The real difference with this integral from the standard “${\int }_a^{b}f(x)dx$” we used in the past is that of context. Our previous integrals naturally summed up values over an interval on the $x$-axis, whereas now we are summing up values over a curve. If we can parametrize the curve with the arc-length parameter, we can evaluate the line integral just as before. Unfortunately, parametrizing a curve in terms of the arc-length parameter is usually very difficult, so we must develop a method of evaluating line integrals using a different parametrization.

Given a curve $C\text{,}$ find any parametrization of $C\text{:}$ $x=g(t)$ and $y=h(t)\text{,}$ for continuous functions $g$ and $h\text{,}$ where $a\le t\le b\text{.}$ We can represent this parametrization with a vector-valued function, $\overrightarrow{r}(t)=⟨g(t),h(t)⟩\text{.}$

By the Fundamental Theorem of Calculus, $ds=\Vert \overrightarrow{r}{}^{\prime }(t)\Vert dt\text{.}$ We can substitute the right hand side of this equation for $ds$ in the line integral definition.

We restate this as a theorem, along with its three-dimensional analogue, followed by an example where we finally evaluate an integral and find an area.

To be clear, the first point of Theorem 15.1.4 can be used to find the area under a surface $z=f(x,y)$ and above a curve $C\text{.}$ We will later give an understanding of the line integral when $C$ is a curve in space.

Let’s do an example where we actually compute an area.

We will practice setting up and evaluating a line integral in another example, then find the area described at the beginning of this section.

We now consider the example that introduced this section.

We give one more example of finding area.

Note how in each of the previous examples we are effectively finding “area under a curve”, just as we did when first learning of integration. We have used the phrase “area over a curve $C$ and under a surface,” but that is because of the important role $C$ plays in the integral. The figures show how the curve $C$ defines another curve on the surface $z=f(x,y)\text{,}$ and we are finding the area under that curve.

Many properties of line integrals can be inferred from general integration properties. For instance, if $k$ is a scalar, then ${\int }_{C}kf(s)ds=k{\int }_{C}f(s)ds\text{.}$

This property allows us to evaluate line integrals over some curves $C$ that are not smooth. Note how in Figure 15.1.13.(b) the curve is not smooth at $D\text{,}$ so by our definition of the line integral we cannot evaluate ${\int }_{C}f(s)ds\text{.}$ However, one can evaluate line integrals over $C_1$ and $C_2$ and their sum will be the desired quantity.

A curve $C$ that is composed of two or more smooth curves is said to be piecewise smooth. In this chapter, any statement that is made about smooth curves also holds for piecewise smooth curves.

We state these properties as a theorem.

We first learned integration as a method to find area under a curve, then later used integration to compute a variety of other quantities, such as arc length, volume, force, etc. In this section, we also introduced line integrals as a method to find area under a curve, and now we explore one more application.

By taking the limit as the length of the segments approaches 0, we have the definition of the line integral as seen in Definition 15.1.3. When learning of the line integral, we let $f(s)$ represent a height; now we let $f(s)=\delta (s)$ represent a density.

We can extend this understanding of computing mass to also compute the center of mass of a thin wire. (As a reminder, the center of mass can be a useful piece of information as objects rotate about that center.) We give the relevant formulas in the next definition, followed by an example. Note the similarities between this definition and Definition 14.6.30, which gives similar properties of solids in space.

We end this section with a callback to the Integration Review that preceded this section. A line integral looks like: ${\int }_{C}f(s)ds\text{.}$ As stated before the definition of the line integral, this means “sum up, along a curve $C\text{,}$ function values $f(s)$ × small arc lengths.” When $f(s)$ represents a height, we have “height × length = area.” When $f(s)$ is a density (and we use $\delta (s)$ by convention), we have “density (mass per unit length) × length = mass.”

In the next section, we investigate a new mathematical object, the vector field. The remaining sections of this chapter are devoted to understanding integration in the context of vector fields.