Use cylindrical shells to find the volume V of a right circular cone with height 7h and base radius 4r.

To find the volume of a right circular cone using cylindrical shells, we can imagine the cone as a stack of infinite number of thin and hollow cylinders (cylindrical shells). Each of these cylindrical shells has a small height and can be representative of a small slice of the cone.

First, let's visualize the right circular cone with height 7h and base radius 4r. The cone can be divided into many small cylindrical shells stacked on top of each other vertically. Let's consider one of these cylindrical shells, which has a height dh and a radius r at that height.

Now, let's find the volume of this cylindrical shell. The volume of a cylindrical shell is given by the formula:

V_shell = 2πrh * dh

Here, 2πrh represents the curved surface area of the cylindrical shell, and dh represents the height (or thickness) of the shell.

We want to find the volume of the entire cone, so we need to add up the volumes of all these infinitely thin cylindrical shells. To do this, we integrate the formula for the volume of each shell from the base of the cone to the top.

V = ∫(2πrh * dh) from 0 to 7h

The integral will give us the total volume of the cone.

Now let's solve the integral:

V = ∫(2πrh * dh) from 0 to 7h
= 2πr * ∫(h * dh) from 0 to 7h
= 2πr * [h^2/2] from 0 to 7h
= 2πr * [(7h)^2/2 - 0^2/2]
= 2πr * [49h^2/2]
= 49πr * h^2

Therefore, the volume of the right circular cone with height 7h and base radius 4r is 49πr * h^2.

I don't see why you need calculus for this, nor why they are calling the height 7h and the radius 4r.

The differential volume of any cylindrical shell of radius r, height h and thickness dr is

2 pi r h dr

Integrating that from r = 0 to the outside radius R will give you the volume.