Dick said:Why do you think the upper limit of the dz integration is r*cos(theta)? Don't you have z=sqrt(a^2-y^2)=sqrt(a^2-r^2*sin(theta)^2)?
Dick said:That's only true along the curve where the two cylinders intersect. It's not true everywhere on the surface in the first octant.
The formula for finding the volume of the intersection of two cylinders using polar coordinates is V = ∫θ1θ2 ∫r1r2 h(r,θ) dr dθ, where θ1 and θ2 are the limits of integration for the angle θ, r1 and r2 are the limits of integration for the radius r, and h(r,θ) is the height of the intersection at a given point (r,θ).
The limits of integration for θ and r can be determined by finding the points of intersection between the two cylinders and using those points to create the boundaries of the integration. These points can be found by setting the equations of the cylinders equal to each other and solving for the values of θ and r.
Yes, the formula can be applied to any two cylinders as long as they have a circular base and their axes are parallel. However, the integration may be more complex for certain cylinder configurations.
The volume of the intersection of two cylinders will vary depending on the angle of intersection. As the angle increases, the volume of the intersection will also increase, reaching its maximum when the two cylinders are perpendicular to each other.
Calculating the volume of the intersection of two cylinders using polar coordinates is important in various fields of engineering and science. It can be used to determine the capacity of cylindrical tanks or pipes, or to analyze the volume of complex geometric shapes for mathematical or scientific purposes.