The height (in meters) of large mountain on Mars can the be accurately predicted by the function h(x,y) where x is the east-west direction (in km) while y is the north-south direction (in km).

The mountain's volume will be analyzed using calculus.

1) The volume can be roughly estimated using the volume of a right cone where:

So, the volume should be well more than 1,000 km³.

2) Excel estimates the volume at about 1756.17 km³ using 1 km intervals.

3) Next. lets get an answer for the volume using calculus (shown on the next page.)

The height function (in meters)

Just for kicks and to help visualize the problem, take the original function (at z = 0) and complete the square (twice) yielding an ellipse with a center at h = 20 and k = 18

at z = 0, solving for y = f(x) yields 2 functions:

solving numerically for the real limits of the ellipse in both the x and y direction with Mathcad's root finding function yields:

setting the graphing domain using these limits (and later used as the integration limits for x):

Integrating (numerically) using the x and y limits with Mathcad and remembering the function z provides results in meters (not km) yields:

(km³)

The Excel answer compares well to this numerical solution.

Integrating this analytically would be laborious. Try it.