Please answer all parts and fully explain for 5 stars! Gabriel\'s Horn All parts
ID: 3344511 • Letter: P
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Please answer all parts and fully explain for 5 stars!
Gabriel's Horn All parts of this problem are based on the function f(x) =1/x2/3 Find an expression for the area between f(x) and the x-axis for x = 1 to x = for some constant > 1. Take the limit as rightarrow infinity of your answer to (1) to find the area under the curve f(x) for x- values from 1 to infinity. Find the volume of the solid formed by rotating the region bounded by f(x) and the x-axis around the x-axis for x-values 1 x , for some constant > 1. Take the limit as rightarrow infinity of your answer to (3) to find the volume of this infinitely long "trumpet" (known as Gabriel's Horn). How can your results to (2) and (4) both be true? Explain.Explanation / Answer
a) integral (1,?) of 1/(x^2/3)dx; ?>1 b) Find area under f(x) from 1 to infinity = Indefinite integral = lim as ? approaches infinity of the integral from 1 to ? of f(x)dx = lim ?->infinity of integral (1,?) of 1/(x^2/3)dx = lim ?->infinity of integral (1,?) of (x^-2/3)dx = lim ?->infinity [3(x^1/3)] from 1 to ? = lim ?->inf [3(?^1/3) - 3(1^1/3)] = lim ?->inf [3(?^1/3 - 3(1)] = lim ?->inf [3(?^1/3)] - lim ?->inf [3] = ( lim ?->inf [3(?^1/3)] ) - 3 = [3(inf^1/3)] - 3 = 3(inf) - 3 = infinity - 3 = infinity = the area is divergent (not a defined value) c) Volume = integral (1, infinity) of pi*(r^2)dx (this is volume by disk method) (r is the radius of each disk = f(x) - axis of rotation = f(x) - 0 = f(x). Thus r^2 is (f(x))^2 = (x^-2/3)^2 =x^-4/3.) = integral (1,?) of pi*(x^-4/3)dx = [pi*(-3)(x^-1/3)] from 1 to ? = -3pi [(?^-1/3) - (1^-1/3)] = -3pi(?^-1/3) + 3pi(1) = 3pi - 3pi(?^-1/3) d) lim ?->inf [3pi - 3pi(?^-1/3)] = 3pi - lim ?->inf [3pi(?^-1/3)] = 3pi - (3pi * infinity^-1/3) = 3pi - (1/infinity) = 3pi - 0 = 3pi d) Gabriel's horn is paradoxical in that it has infinite surface area but finite volume. Thus, the answers can both be true. You can fill this "horn" with a finite number of units (in this case, 3pi cubic units), but it would take infinitely many units to cover its surface area.
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