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Volume

Volume. Claudine Cho Period 1 November 9, 2009 . Washer Method. Find the volume of y= X^2, y=4 revolved around the x-axis Cross sections are circular washers Thickness of the washer is xsub2-xsub1 Step 1) Find the thickness.

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Volume

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  1. Volume Claudine Cho Period 1 November 9, 2009

  2. Washer Method • Find the volume of y= X^2, y=4 revolved around the x-axis • Cross sections are circular washers • Thickness of the washer is xsub2-xsub1 • Step 1) Find the thickness. (Volume of entire)^2 – (Volume of the “hole”)^2. In this case it is (4)^2-(X^2)^2

  3. Washer Cont. • Step 2) Find the bounds. Because y=4 and y=x^2 intersect at x=-2 and x=2, those are your bounds. • Step 3) Plug all the information into the integral • It should like this : • from x=-2 to x=2 of (4)^2-(X^2)^2 all multiplied by pie • The answer should be : 256pie/5

  4. Disk Method • Find the volume of y=X^2 between x=0 and x=1 revolved around the x-axis • Step 1) For this problem, you must do “top-bottom” to find the “thickness”, so it is ((x^2)-0) • Step 2) Next, you must find the bounds. As given in the problem, the bounds are from x=0 to x=1

  5. Disk Cont. • Step 3) Set up the integral from x=0 to x=1 and square the thickness of the disk, which is ((x^2)-0) • In every disk problem, you will need to square the “thickness” • Step 4) multiply everything by pie. • It should look like this: • from x=0 to x=1, ((x^2)-0)^2 dx all multiplied by pie • The answer should be: pie/5

  6. Shell Method • Find the volume of the solid generated when the curve y=x^2 between x=1, x=2 is rotated around the y-axis • Step 1) Sketch a line segment parallel to axis of revolution. This is the height of a cylinder.

  7. Shell Method Continued • Step 2) Connect this segment perpendicular to the axis of revolution. This is the radius of a cylinder. • Step 3)Find the limits of integration. If rotating around x-axis, then it will be dy. If rotating around y-axis, then it will be dx. • Step 4) Integrate using Surface Area= (2)(r)(pie)

  8. Cont. • So, it would be: 2pie from 1 to 2 of (radius)(height) dx radius=(x) & height=(x^2-0) The answer should be :15pie/2

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