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Calculus II Chapter 6 Section 4 Volume using Shells

Calculus II Chapter 6 Section 4 Volume using Shells. David Dippel LoneStar College - Montgomery. Revolve the region bounded by y = x 2 +1 y = 0 and x = 0 around the y-axis. cross section. If we take a vertical slice. and revolve it about the y-axis. we get a cylinder.

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Calculus II Chapter 6 Section 4 Volume using Shells

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  1. Calculus IIChapter 6 Section 4Volume using Shells David Dippel LoneStar College - Montgomery

  2. Revolve the region bounded by y = x2+1 y = 0 and x = 0 around the y-axis. cross section If we take a vertical slice and revolve it about the y-axis we get a cylinder. If we add all of the cylinders together, we can reconstruct the original object.

  3. cross section The volume of a thin, hollow cylinder is given by: r is the x value of the function. h is the y value of the function. thickness is dx.

  4. This is called the shell method because we use cylindrical shells. cross section If we add all the cylinders from the smallest to the largest:

  5. Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

  6. If we take a vertical slice and revolve it about the y-axis we get a cylinder. Shell method:

  7. Note: When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.

  8. When the strip is parallel to the axis of rotation, use the shells. When the strip is perpendicular to the axis of rotation, use disks. Long side of Rectangle Short Side of Rectangle Distance to Rectangle Long side of Rectangle Short Side of Rectangle p

  9. Find the volume of the region generated by rotation y = 2x, y = 4, and x = 0 around:the x-axisthe line y = 5

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