Volume+by+Slices

Volumes by Slices Definition: The volume of a solid of a known integrable cross-section area A (x) from x = a to x = b is the integral of A from a to b: (source [|http://faculty.eicc.edu/bwood/ma][|supnotes/suppleme][|th150][|ntal25.htm])

Volumes by slices allows us to approximate the volume of a 3D shape using cross-section areas. To make the approximation more accurate, use more slices and thinner slices.

In order to find the volume of a shape in between two graphs, follow the steps below. 1) Draw the graph 2) Find the limits of integration of the solid 3) Integrate the equation

ex: Find the volume of the object whose base is bounded by the y-axis and y=4 and whose cross sections are squares.


 * 1) Draw the graph**:

The limits of integration are from 0 to 4
 * 2) Find the limits of integration of the solid:**

The equation is squared because we are using square cross sections and the area for a square is or just (This equation would be different if you were using different shapes for the cross sections such as semi circles or right triangles). To make this volume instead of area, you must multiply and here, the height is dx.
 * 3) Integrate the equation**:

(Not drawn to scale)

The graph above shows the graph of with square cross sections. To solve this integral, you must plug in 4 and 0 for x and subtract the two:



For finding volumes by slices for semi circles, instead of using the formula, you would use:

For finding volumes by slices for right triangles, instead of using the formula, you would use:

Find the volume of the object whose base is represented by the equation bounded by the the y-axis and x=7 and whose cross sections are semi circles.
 * Practice Problem:**

Answer: