Line Integrals in the Plane

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.