Chapter 7.2: Volume The Disk Method The Washer Method Cross-sections 2014 - 2015

Finding Volume Goes back to our first attempt at finding area under the curve.

Things to figure out What happens if we make our rectangles (disks) infinitely small? How do you find the volume of a disk? (is there another name for disk?) What would the radius be?

The Disk Method Formula

Example 1: Find the volume of the solid formed by revolving the region bounded by the graph of 𝑓 𝑥 = sin 𝑥 and the x-axis 0≤𝑥≤𝜋 about the x-axis.

Example 2: Find the volume of the solid formed by the function 𝑓 𝑥 = 9− 𝑥 2 and the x-axis 0≤𝑥≤3 about the x-axis.

Example 3: Find the volume of the solid formed by the function 𝑦=4− 𝑥 2 4 and the axis 𝑦=2 −3≤𝑥≤3 about the axis 𝑦=2

The Washer Method What if the region looks like this??

Washer Method Formula

Example 1: Find the volume of the solid formed by revolving the region bounded by the graphs of 𝑦= 𝑥 𝑎𝑛𝑑 𝑦= 𝑥 2 about the x-axis.

Example 2: Find the volume of the solid bounded by the functions 𝑦= 𝑥 2 𝑎𝑛𝑑 𝑦= 𝑥 5 about the x-axis.

Find the volume of the solid formed by revolving the region bounded by the graphs of 𝑦= 𝑥 2 +1, 𝑦=0, 𝑥=0, 𝑎𝑛𝑑 𝑥=1 about the y-axis.