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VOLUME BY DISK or disc

VOLUME BY DISK or disc. BY: Nicole Cavalier & Alex Nuss. Volumes. To find the volume of a solid S: Divide S into n “slabs” of equal width Δ x (think of slicing a loaf of bread) the sum of the cylinder areas is a good approximation for the volume of the solid

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VOLUME BY DISK or disc

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  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→∞. x Let S be a solid that lies between x=a and x=b. If the cross-sectional area of S in the plane Px, perpendicular to the x-axis, is A(x), where A is an integrable function, then the volume of S is

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

  4. Example of a disk The volume of each disk is: 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. In this case: r= the y value of the function thickness = a small change in x =dx

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

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

  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=3x2+24x+5 *if you do this, you receive a special prize. 

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