Two Axis
Find the volume between two lines when rotated about the x axis using the cylindrical method?
(BC Calculus) Find the volume between two lines when rotated about the x axis using the cylindrical method (integration). Between the lines x = (y ^ 2 / 2) x = y ^ 4 / 4 – y ^ 2 / 2 and the range of 0-2, and both lines are in (2.2) and (0.0) Please show me the process. Thank you.
First, Graph the two equations. Then draw in a typical cross section between the two lines. Here, the cross section of thickness d is (Since they are rotating around the axis x). In addition, the radius or distance from the axis of rotation of the cross section is the variable (o-0 and technically). The height (or length, however, you choose to look at it) of this section is the difference of the two functions. Subtracting take over the function which is more positive over shaft is spinning you around, in this case the x-axis After graphics, it seems that y ^ 2 / 2 is the background function and ^ y 4 / 4 – y ^ 2 / 2 is the function superior. And of course along the boundaries and range from 0-2. Therefore, we can now set our integral: 2 * pi * [int (y = 0 and y = 2) y ^ 2 / 2 - (^ 4/4-y ^ 2 / 2) *] dy = 2 * pi * [int (y = 0 y = 2) (-y ^ 4 / 4 + y ^ 2) *] dy = 2 * pi * [(5.20-y ^ + y ^ 3 / 3) eval @ y = 2 & Y = 0] = 2 * pi * [(-2 ^ 5 / 20 +2 ^ 3 / 3)] = 2 * pi * [8 / 3-8 / 5] = 2 * pi * 40 / 15 [- 24 / 15] = 32 * pi/15 Viola!
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