In general integrals in spherical coordinates will have limits that depend on the 1 or 2 of the variables. The pilot has very little control over where the balloon goes, however—balloons are at the mercy of the winds.
A region bounded below by a cone and above by a sphere. In this project we use triple integrals to learn more about hot air balloons.
The heat is generated by a propane burner suspended below the opening of the basket. A graph of our balloon model and a cross-sectional diagram showing the dimensions are shown in the following figure.
Find the volume of the balloon in two ways. Let us look at some examples before we consider triple integrals in spherical coordinates on general spherical regions. In these cases the order of integration does matter.
The length in the r and z directions is dr and dz, respectively. For some problems one must integrate with respect to r or theta first.
The top of the balloon is modeled by a half sphere of radius 28 feet. Triple Integrals in Cylindrical Coordinates Cylindrical coordinates are obtained from Cartesian coordinates by replacing the x and y coordinates with polar coordinates r and theta and leaving the z coordinate unchanged.