The volume generated by rotating, about the X𝑋 axis, the region enclosed by y=x32𝑦=𝑥32, x=1,x=2𝑥=1,𝑥=2, and the X𝑋 axis, is Answer 1 Question 9

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The volume generated by rotating, about the X𝑋 axis, the region enclosed by

$y = x^{3/2}$ ,
$x = 1$ ,
$x = 2$ ,
and the X𝑋 axis, is

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Solution

To find the volume generated by rotating the region enclosed by y=x^(3/2), x=1, x=2, and the x-axis, we use the disk method. The disk method formula is V = π ∫ [R(x)]^2 dx from a to b, where R(x) is the radius of the disk at x and [a, b] is the interval over which we're rotating.

Here, R(x) = x^(3/2), a = 1, and b = 2.

So, we have:

V = π ∫ from 1 to 2 of [x^(3/2)]^2 dx = π ∫ from 1 to 2 of x^3 dx = π [ (1/4)x^4 ] from 1 to 2 = π [(1/4)(2^4) - (1/4)(1^4)] = π [(1/4)(16) - (1/4)(1)] = π [4 - 1/4] = π [15/4] = (15π/4) cubic units

So, the volume of the solid generated is (15π/4) cubic units.

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The volume generated by rotating, about the X𝑋 axis, the region enclosed by y=x32𝑦=𝑥32, x=1,x=2𝑥=1,𝑥=2, and the X𝑋 axis, is Answer 1 Question 9

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