Consider the solid shaped like an ice cream cone that is bounded by the functions z=x^2+y^2 and z=18−x^2−y^2. Set up an integral in polar coordinates to find the volume of this ice cream cone.

Answer 1

Let z= x² +y² and z=18-x² -y² then equating these two functions gives x² + y² =9 and r=3 so 0≤r≤3 and

r²≤z≤ 18-r² the integral becomes ∫₀^2π∫₀³∫_r^2^{18-r^2} dzrdrdθ. which can be solved directly.