Gabriel’s Horn All parts of this problem are based on the function f(x) = 1/x^2/3.
(1) Find an expression for the area between f(x) and the x-axis for x = 1 to x = ☺ for some constant ☺ > 1.
(2) Take the limit as ☺ --> ∞ of your answer to (1) to find the area under the curve f(x) for x-values from 1 to ∞.
(3) 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.
(4) Take the limit as ☹ --> ∞ of your answer to (3) to find the volume of this infinitely long “trumpet” (known as Gabriel’s Horn).
(5) How can your results to (2) and (4) both be true? Explain.
I solved to following as below, my question is how did I end up with a negative volume for #4?
(1) Find an expression for the area between f(x) and the x-axis for x = 1 to x = ☺ for some constant ☺ > 1.
(2) Take the limit as ☺ --> ∞ of your answer to (1) to find the area under the curve f(x) for x-values from 1 to ∞.
(3) 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.
(4) Take the limit as ☹ --> ∞ of your answer to (3) to find the volume of this infinitely long “trumpet” (known as Gabriel’s Horn).
(5) How can your results to (2) and (4) both be true? Explain.
I solved to following as below, my question is how did I end up with a negative volume for #4?