Calculus Notes Ch 6.2 'Volumes by slicing can be found by adding up each slice of the solid as the thickness of the smaller and smaller, in other words the number of slices goes to infinity. 'To find the volume of each slice of the pyramid you would find the area of each square then multiply the area by the thickness of the slice. The thickness would be dx because we are slicing with respect to the x-axis. Next you would add the volumes of each slice together to find the total volume. 'This would be a Riemann sum with the limit as n the number of slices going to infinity.
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Remember the from Ch 5.5 A = lim E f (xi)Ah = nux Ax—O k —l Only now instead of f(x*) we will be summing the areas of the base of each slice. So change f(x*)in the Riemann Sum to A(x*)which represents the area Of the base. b V = lim = nux k=l
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with respect to the x-axis. A = Tcr2 A = 1122 = 411 f 4m dx 2 — 162 = 41'T(6) — Here is an example using a 4 cylinder Find the volume of the cylinder using the formula and slicing — 8m = 16m 2 6 41T(2)
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Now use the formula you learned in Geometry to find the area ' V=lt r2 = 16 It You can also work problems like these with respect to the y-axis. 6
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Volume of solids of revolution using disks If you take the area under the line y = x belOWwill form a cone (see below diagram)
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Now use the formula below to find the volume of the 3-D figure formed by rotating around the x-axis. http://www.plu.edu/—heathdj/java/calc2/Solid.html This method is called disks when revolved around the x-axis ( note: it is sliced with respect to the x-axis and is revolved around the x-axis)
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Formula for finding the volume of the solid formed when f(x) is revolved around the x- axis Each slice is a circle, the formula for the area of a circle is A-it r2. The radius is the y- value of the function AKA f(x). So, the area formula becomes A-it (f(x))2 Substitute into the formula b b A(x) dx = f IT (f a a
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Equation y = x from 0 to 4 With geometry V = 1/3 It With Calculus f IT (X2) dx 3 (634 0) 3
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Your turn p. 410 # 6. find the volume of the solid that results when the region enclosed by the given curves is revolved around the x-axis Y = sec x, x = — tan IT(f Use the formula IT(sec x)2dx = IT[tan x 3—1 = n tan 3
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