The Gateway Arch in St. Louis (see the photo on page 463)...

The Gateway Arch in St. Louis (see the photo on page 463) was constructed using the equation $$ y=211.49-20.96 \cosh 0.03291765 x $$ for the central curve of the arch, where $x$ and $y$ are measured in meters and $|x| \leqslant 91.20$. Set up an integral for the length of the arch and use your calculator to estimate the length correct to the nearest meter.

The Gateway Arch in St. Louis (see the photo on page 463) was constructed using the equation

$$
y=211.49-20.96 \cosh 0.03291765 x
$$

for the central curve of the arch, where $x$ and $y$ are measured in meters and $|x| \leqslant 91.20$. Set up an integral for the length of the arch and use your calculator to estimate the length correct to the nearest meter.

Key Concepts

Arc Length of a Curve

The arc length of a curve defined by a function y = f(x) is given by the integral L = ? sqrt{1 + (dy/dx)²} dx. This formula accounts for both horizontal and vertical changes along the curve and is derived from the Pythagorean theorem applied to infinitesimally small segments of the curve.

Differentiation of Hyperbolic Functions

When dealing with curves that involve hyperbolic functions such as cosh and sinh, it is essential to differentiate these functions correctly. The derivative of the hyperbolic cosine function is the hyperbolic sine function scaled by any constant factor present in the argument, which then contributes to the arc length integrand through the term (dy/dx)².

Numerical Integration

Often, the integral used to find arc length cannot be expressed in a simple closed form. In these cases, numerical techniques—such as using a calculator or numerical approximation methods like Simpson’s rule or the trapezoidal rule—are employed to estimate the value of the integral to the required accuracy.

Symmetry in Integration

If the function whose arc length is being computed exhibits symmetry, the integration can be simplified by calculating the arc length over a symmetric subinterval and then doubling the result. This approach reduces the computational effort while still providing an accurate measurement of the full length.

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