This video demonstrates how to calculate the volume enclosed between a cone and a sphere.
First, I explain how to find the region of intersection by equating the equations of the two surfaces. This reveals that the intersection is a circle with a radius of 2√2. I then describe the conversion to polar coordinates using x = r cos θ and y = r sin θ.
Next, I outline the volume calculation process. The volume is found by double integration over the region R, summing infinitesimally small volume elements. Each element is the product of an infinitesimal area (dA) and a height function f(x, y).
I then detail the integration process. I explain how the infinitesimal area element dA in polar coordinates becomes r dr dθ, where dθ is the infinitesimal angle of a circular sector. The inner integral integrates along the r direction, and the outer integral integrates as r rotates through θ from 0 to 2π.
To determine the height function f(x, y), I subtract the z-value of the lower surface (cone) from the z-value of the upper surface (sphere). I explain how setting x and y to zero can help determine which function is "on top." This height function is then converted to polar coordinates using x = r cos θ and y = r sin θ.
Finally, I evaluate the inner integral (with respect to r) and then the outer integral (with respect to θ) to arrive at the solution for the enclosed volume.