Integral Calculus 4

Here is another shape to find the volume for. The shape has a base which is a disk of radius 1, and each cross section is an equilateral triangle.

View of the base

Top view

Integral Calculus 3

Here is another shape for which calculus allows us to compute volume. It is a shape for which the base is the region bounded by y=4-x^2 and the x-axis

and every vertical cross section is a square:

Integral Calculus 2

This object illustrates the disk and shell methods for finding volumes of rotation using integration. Take the region bounded by the graph of the function f(x) = -x^2 - x + 8 and the x-axis from x=0 to x=2:

and rotate it about the y-axis. In the middle is the resulting solid of revolution 

On one side is the approximation of the volume using four disks, and on the other side is the approximation of the volume using four shells. To make it clear which is which, I have shown the same model with the pieces taken apart.

 This model is "Volumes of Hanoi" designed by mathgrrl.

Integral Calculus

I am preparing some integral calculus prints. The first is the solid obtained by revolving the region between the graph of y=4-x^2 and the x-axis on [0,2] around the y-axis, and the model on the right is an approximation of that solid using eight shells, designed by mathgrl.

Here is a view of all the calculus volumes collection. 

Four volumes

Pencil for scale