Find the volume of a right circular cone of height h and semi-vertical angle a
(Hint: Rotate about the x-axis).
r/h = sin(a)
v = 1/3 pi r^2 h = 1/3 pi h^3 sin(a)
To find the volume of a right circular cone, we need to consider the shape formed by rotating a right triangle around the x-axis.
Let's assume the radius of the base of the cone is denoted by r.
Now, when we rotate the right triangle around the x-axis, we form a cone. The base of the cone is a circle with radius r, and the height of the cone is h.
The semi-vertical angle, a, is the angle between the slant height of the cone and the height of the cone. So, we can consider the right triangle formed by the height, slant height, and radius of the base.
Using trigonometry, we can relate the height, slant height, and the semi-vertical angle as follows:
sin(a) = r / l
Where l is the slant height of the cone.
From this equation, we can express r in terms of l and a:
r = l * sin(a)
Now, to find the volume of the cone, we can integrate the cross-sectional area as we rotate the right triangle around the x-axis.
The cross-sectional area of each slice of the cone, at a particular x-coordinate, is given by the formula:
A(x) = π * r^2
Since r is a function of l, we need to express A(x) in terms of l.
Using the relation r = l * sin(a), we find:
A(x) = π * (l * sin(a))^2
= π * l^2 * sin(a)^2
To find the volume, we need to integrate A(x) over the range of x from 0 to h (the height of the cone):
V = ∫[0 to h] A(x) dx
= ∫[0 to h] (π * l^2 * sin(a)^2) dx
After integrating and simplifying, we get the volume of the right circular cone as:
V = (π * h * l^2 * sin(a)^2) / 3
Therefore, the volume of a right circular cone of height h and semi-vertical angle a, when rotated about the x-axis, is (π * h * l^2 * sin(a)^2) / 3.