Unformatted text preview: Volume by Slicing (Section 6.1) In general, the volume of a cylindrical solid is: What if we have a more complicated object? a xi ‐ 1 xi b The volume of the ith slice is approximately: Vi = Then the volume of the whole solid is estimated by the sum: This is a Riemann sum so we can take the limit of the sum as n → ∞ to get the exact value of the volume. n b i =1 a V = lim ∑ A( xi )Δxi = ∫ A( x ) dx n→ ∞ where A( x ) is a function representing the cross‐sectional area at x. Calculating the volume of a solid by slicing: 1) 2) 3) 4) Example 1: The base of the solid is the circle x 2 + y 2 = 1 . Cross sections are perpendicular to the x – axis and are semi‐circular disks with diameters in the xy – plane. Example 2: The base of the solid is the circle x 2 + y 2 = 1 . Cross sections are perpendicular to the x – axis and are triangles with bases in the xy – plane. Example 3: A solid has slices perpendicular to the y – axis that are squares with one edge in the xy – plane. The intersection of the solid with the xy – plane is the region between the curves x = y 3 − 9 and x = −2 y 2 − 9 . Find the volume of the solid. Example 4: Let R be the region bounded by y = 3, y = x 3 , and x = 0. Find the volume of the solid with base R and cross‐sections perpendicular to the y – axis that are equilateral triangles. ...
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This note was uploaded on 10/02/2011 for the course MATH 2214 at Virginia Tech.