smc-1180-70-Partial Fractions

Part of the graph of `y = (2)/(sqrt(x^2-4x+3))`, where `x > 3`, is shown below.

Find the volume of the solid of revolution formed when the graph of `y = (2)/(sqrt(x^2-4x+3))` from `x = 4` to `x = 6` is rotated about the `x`-axis. Give your answer in the form `a log_e(b)` where `a` and `b` are real numbers. (5 marks)

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`pi log_e ((9)/(5))`

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`V = pi int_4 ^6 (4)/(x^2 – 4x + 3)\ dx`

`text(Using partial fractions:)`

`(4)/(x^2 – 4x + 3) = (a)/((x-3)) + (b)/((x-1))`

`a(x -1) + b(x – 3)= 4`

`text(When)\ \ x = 1, \ -2b = 4 => \ b = -2`

`text(When)\ \ x = 3, \ 2a = 4 => \ a = 2`

`:. \ V`	`= pi int_4 ^6 (2)/(x-3) – (2)/(x-1)\ dx`
	`= 2 pi [log_e(x-3) -log_e(x-1)]_4 ^6`
	`= 2 pi [log_e 3 – log_e 5 – (log_e 1 – log_e3)]`
	`= 2 pi (2log_e 3 – log_e 5)`
	`= 2pi log_e((9)/(5))`