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Greg Kelly, Hanford High School, Richland, Washington Adapted by: Jon Bannon, Siena College Photo by Vickie Kelly, 2006 8.2-8.3 Day 3 The Shell Method and Arc length Japanese Spider Crab Georgia Aquarium, Atlanta
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Find the volume of the region bounded by,, and revolved about the y - axis. We can use the washer method if we split it into two parts: outer radius inner radius thickness of slice cylinder Japanese Spider Crab Georgia Aquarium, Atlanta
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If we take a vertical sliceand revolve it about the y-axis we get a cylinder. cross section If we add all of the cylinders together, we can reconstruct the original object. Here is another way we could approach this problem:
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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.
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cross section If we add all the cylinders from the smallest to the largest: This is called the shell method because we use cylindrical shells.
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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.
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Shell method: If we take a vertical slice and revolve it about the y-axis we get a cylinder.
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Note:When entering this into the calculator, be sure to enter the multiplication symbol before the parenthesis.
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When the strip is parallel to the axis of rotation, use the shell method. When the strip is perpendicular to the axis of rotation, use the washer method.
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If we want to approximate the length of a curve, over a short distance we could measure a straight line. By the pythagorean theorem: We need to get dx out from under the radical. Length of Curve (Cartesian)Lengths of Curves:
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Example: Now what? This doesn’t fit any formula, and we started with a pretty simple example! The TI-89 gets:
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Example: The curve should be a little longer than the straight line, so our answer seems reasonable. If we check the length of a straight line:
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Example: You may want to let the calculator find the derivative too: Important: You must delete the variable y when you are done! ENTER F44 Y STO Y
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Example:
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If you have an equation that is easier to solve for x than for y, the length of the curve can be found the same way. Notice that x and y are reversed. ENTER X STO
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Don’t forget to clear the x and y variables when you are done! ENTER F44 Y X
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