15.3

Line Integrals in the Plane

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A mural is planned for the curved outer wall of a clock monument. To find the space available for the artwork, the total surface area of the curved wall must be found.

The base of the wall follows a curved path, C. This curve is defined by parametric equations, where x and y depend on the parameter t. As t changes from a to b, it traces the curve from its starting point to its ending point.

The height of the wall also changes from point to point along the curve and is given by a function of x and y.

To find the total surface area, the wall is divided into n vertical strips along its curved base.

Each strip is treated as a thin rectangle, with a width equal to a small arc length and a height equal to the wall’s elevation. Multiplying these values gives the area of each small rectangle.

Adding these small areas forms a Riemann sum. As the widths of these strips become infinitesimally small, the sum converges to a line integral.

Evaluating this line integral using the curve length and the height function gives the total surface area of the wall. This gives the space available for the mural on the outer wall of the clock monument.

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Line integrals in the plane provide a method for evaluating quantities distributed along a curve, such as mass, work, or surface area. A curve C in the plane is commonly represented parametrically by x = x(t) and y = y(t), where the parameter t varies over an interval [a, b]. This representation allows geometric and physical quantities to be expressed in terms of a single variable, facilitating both analysis and computation.

A line integral of a scalar function f(x, y) along a curve C is defined as the limit of a Riemann sum. The curve is partitioned into small segments, each with length Δs. Over each segment, the function is approximately constant, so the contribution is f(x_i,y_i)Δs_i. Summing these contributions and taking the limit yields

\begin{equation*}\int_C f(x,y)\,ds\end{equation*}

This integral represents the accumulation of the function f along the curve, weighted by arc length. Geometrically, it can describe quantities such as the mass of a wire with variable density or the area of a vertical surface with varying height.

When the curve is parameterized, the arc length element is expressed as

\begin{equation*}ds=\bm{\sqrt{\Biggl(\jfrac{\textit{d}\textit{x}}{\textit{d}\textit{t}}\Biggr)^2+\Biggl(\jfrac{\textit{d}\textit{y}}{\textit{d}\textit{t}}\Biggr)^2}}\,\textit{d}\textit{t}\end{equation*}

Substituting into the line integral gives

\begin{equation*}\int_C \textit{f}(\textit{x},\textit{y})\,ds=\int_a^b \textit{f}(\textit{x}(\textit{t}),\textit{y}(\textit{t}))\bm{\sqrt{\Biggl(\jfrac{\textit{dx}}{\textit{dt}}\Biggr)^2+\Biggl(\jfrac{\textit{dy}}{\textit{dt}}\Biggr)^2}}\,\textit{dt}\end{equation*}

This formulation highlights how both the function and the curve's geometry influence the result.

Line integrals are widely used in applications involving distributed quantities. For example, in determining the surface area of a curved wall, the height function plays the role of f(x,y), while the curve defines the base. Similarly, in physics, line integrals describe work done by a force field along a path or the mass of a non-uniform wire.

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