Section 15.8: Triple Integration in Spherical Coordinates

The latitude and longitude of a point P in the Northern Hemisphere are related to spherical coordinates \(\rho, \theta, \phi\) as follows. We take the origin to be the center of the earth and the positive z-axis to pass through the North Pole. The positive x-axis passes through the point where the prime meridian (the meridian through Greenwich, England) intersects the equator. Then the latitude of P is \(\alpha = 90^\circ - \phi^\circ\) and the longitude is \(\beta = 360^\circ - \theta^\circ\). Find the great-circle distance from Los Angeles (lat. 34.06 N, long. 118.25 W) to Montreal (lat. 45.50 N, long. 73.60 W). Take the radius of the earth to be 3960 mi. (A great circle is the circle of intersection of a sphere and a plane through the center of the sphere.)