1. VOLUME BY DISK or disc BY: Nicole Cavalier & Alex Nuss

2. Volumes To find the volume of a solid S: Divide S into n “slabs” of equal width Δx (think of slicing a loaf of bread) the sum of the cylinder areas is a good approximation for the volume of the solid the approximation is getting better as n→∞. Let S be a solid that lies between x=a and x=b. If the cross-sectional area of S in the plane P x, perpendicular to the x-axis, is A(x), where A is an integrable function, then the volume of S is x

3. VOLUME BY DISK If the shape is rotated about the x-axis, then the formula is: A shape rotated about the y-axis would be:

4. How could we find the volume of the cone? One way would be to cut it into a series of disks (flat circular cylinders) and add their volumes. The volume of each disk is: In this case: r= the y value of the function thickness = a small change in x = dx Example of a disk

5. The volume of each flat cylinder (disk) is: If we add the volumes, we get:

6. The region between the curve, and revolved about the y-axis. Find the volume. We use a horizontal disk. The thickness is dy. The radius is the x value of the function. volume of disk Example of rotating the region about y-axis = 4.355

7. The natural draft cooling tower shown at left is about 500 feet high and its shape can be approximated by the graph of this equation revolved about the y-axis: The volume can be calculated using the disk method with a horizontal disk.

8. EVERYONE GET UP AND LETS STAND IN A LINE BY AGE!* y=3x 2 +24x+5 *if you do this, you receive a special prize.