Using the disk method, find the volume of a right circular cone with height h and base radius r. I know the answer is 1/3pier^2h, but how do you explain or work out the problem?
To find the volume of a right circular cone using the disk method, we can break down the cone into infinitesimally thin disks stacked on top of each other. The volume of each disk is approximated using the formula for the volume of a cylinder: V_disk = π * r^2 * Δh, where r is the radius of the disk and Δh is its height.
To determine the radius of each disk, imagine a cross-section of the cone perpendicular to its base. It forms a triangle, with the height h and the base radius r. From this cross-section, we can observe that the radius of each disk is directly proportional to the height of the disk, relative to the height of the entire cone.
Let's assume that the height of each disk is represented by Δy. Using similar triangles between the cross-section and the disk, we can create the following proportion:
(r / h) = (r_disk / Δy)
By rearranging the equation, we get:
r_disk = r / h * Δy
Now, to find the volume of the entire cone, we need to sum up the volumes of all the disks. This involves integrating the volume of each disk with respect to the height of the disk, from 0 to h.
Therefore, the volume of the cone can be expressed as:
V_cone = ∫[from 0 to h] (π * r^2 * Δy)
Now, we can replace r_disk with its equivalent value:
V_cone = ∫[from 0 to h] (π * (r / h * Δy)^2) = ∫[from 0 to h] (π * r^2 / h^2 * Δy^2)
Since Δy is infinitesimally small, we can rewrite Δy^2 as dy, resulting in:
V_cone = ∫[from 0 to h] (π * r^2 / h^2 * dy)
By integrating this expression with respect to y, we get:
V_cone = π * r^2 / h^2 * (y) | from 0 to h
Substituting the limits of integration, we have:
V_cone = π * r^2 / h^2 * (h - 0) = π * r^2 / h^2 * h
Finally, simplifying the expression, we have:
V_cone = π * r^2 * h / h^2 = π * r^2 * (1/h)
And since 1/h is equivalent to h^(-1), we end up with the formula for the volume of a right circular cone: V_cone = (1/3) * π * r^2 * h