Integration question.. please help!
A part of the curve y = a/x, where a is a positive constant, is rotated through 360◦ about the x-axis between x =1 and x=3. Find the value of 'a' if the volume obtained is 24pi.
I don't know how to integrate the curve y = a/x.. please help me solve the question above with an explanation.
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You need to use the equation that gives the volume of solid of revolution, such that:
`V = pi*int_a^b y^2 dx`
You should notice that the problem provides the limits of integration, `a = 1` and `b = 3` , the value of volume `V = 24 pi` and the function `y = a/x` such that:
`24pi = pi*int_1^3 (a/x)^2 dx => 24 = a^2 int_1^3 1/x^2 dx`
`24 = a^2 int_1^3 x^(-2) dx => 24 = -a^2*(1/x)|_1^3`
Using the fundamental formula of calculus yields:
`24 = -a^2*(1/3 - 1/1) => 24 = (2/3)a^2 => a^2 = (3*24)/2`
`a^2 = 36 => a = +-6`
Since the problem specifies that a is a positive constant yields that `a = 6` .
Hence, evaluating the value of positive constant a, under the given conditions, yields `a = 6.`
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