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Aim: How do we find volume using the Shell Method?

Aim: How do we find volume using the Shell Method?. Do Now:. Find the volume of the solid that results when the region bounded by x = y 3 and x = y 2 from y = 0 to y = 1 is revolved about the y -axis. h. h. w. Axis of Revolution. p. p. The Shell Revolution. h. w.

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Aim: How do we find volume using the Shell Method?

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  1. Aim: How do we find volume using the Shell Method? Do Now: Find the volume of the solid that results when the region bounded by x = y3 and x = y2 from y = 0 to y = 1 is revolved about the y-axis.

  2. h h w Axis of Revolution p p The Shell Revolution

  3. h w Outer radius Inner radius The Shell Method

  4. inner radius outer radius The Shell Method take to limit

  5. Horizontal Axis of Revolution Vertical Axis of Revolution h(y) d h(x) c a b p(x) Horizontal Axis of Revolution Vertical Axis of Revolution The Shell Method To find the volume of a solid of revolution with the shell method

  6. Vertical Axis of Revolution = x – x3 h(x) p(x) = x Model Problem Find the volume of the solid of revolution formed by revolving the region bounded by y = x – x3 and the x-axis (0 <x< 1) about the y-axis.

  7. Horizontal Axis of Revolution p(y) h(y) Model Problem Find the volume of the solid of revolution formed by revolving the region bounded by and the y-axis (0 <y< 1) about the x-axis.

  8. r R r R a b d h p c h p b a Comparing Disc and Shell Methods d Disc Method c Shell Method

  9. Aim: How do we find volume using the Shell Method? diameter Do Now: The bases of a solid is the region enclosed by the graphs of y = 1/2x2 and y = 8. Cross sections perpendicular to the y-axis are semicircles with diameter in the plane of the region. Find the volume of the solid.

  10. Do-Now

  11. disk washer Recall: Model Problem washer disk

  12. h(x) p(x) Shell Method is Preferable = x2 + 1 Vertical Axis of Revolution = x

  13. Disk method  a. rev 2 ft. Shell method  a. rev 8 ft. Model Problem A pontoon is to be made and is designed by rotating the graph of about the x-axis, where x and y are measured in feet. Find the volume.

  14. R(x) = Model Problem – Disk Method A pontoon is to be made and is designed by rotating the graph of about the x-axis, where x and y are measured in feet. Find the volume. r(x) = 0

  15. Horizontal Axis of Revolution p(y) = y Model Problem – Shell Method A pontoon is to be made and is designed by rotating the graph of about the x-axis, where x and y are measured in feet. Find the volume. = h(y) 2( ) h(y)

  16. - ( 1) p(x) Model Problem – Only Shell Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, x = 1 about the line x = 2. (1, 3) Axis of Revolution = 2 – x can’t solve for x

  17. Model Problem – Only Shell Find the volume of the solid of revolution formed by revolving the region bounded by the graphs of y = x3 + x + 1, y = 1, x = 1 about the line x = 2. (1, 3) can’t solve for x Axis of Revolution - ( 1) Vertical Axis of Revolution p(x) = 2 – x

  18. The Shell Method

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