How do you find the volume of the solid obtained by rotating the region bounded by the curves $x = y$ $x=y$ and $y = \sqrt{x}$ $y=sqrtx$ about the line $x = 2$ $x=2$ ?

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How do you find the volume of the solid obtained by rotating the region bounded by the curves $x = y$ $x=y$ and $y = \sqrt{x}$ $y=sqrtx$ about the line $x = 2$ $x=2$ ?

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Answer 1 · 2017-08-10T15:44:42

Please see below.

Explanation:

Here is a graph of the region.

enter image source here

I've taken a slice perpendicular to the axis of rotation. The slice is taken at a variable value of $y$ $y$ .

The thickness of the slice is $d y$ $dy$ , so we need the equations in the form $x =$ $x =$ a function of $y$ $y$ .

The curve on the left ( $y = \sqrt{x}$ $y = sqrtx$ ) is $x = y^{2}$ $x = y^2$

on the right is the line $x = y$ $x = y$

Rotating the slice will generate a washer of thickness $d y$ $dy$ and
volume $π (R^{2} - r^{2}) d y$ $pi(R^2-r^2) dy$
where $r$ $r$ is the outer radius and $r$ $r$ the inner.

The outer radius of the washer is the distance between the curve on the left and the line $x = 2$ $x=2$ .
So $R = 2 - y^{2}$ $R = 2-y^2$

The inner radius is the distance from the line on the right and the line $x = 2$ $x=2$ .
So $R = 2 - y$ $R = 2-y$

$y$ $y$ varies from $0$ $0$ to $1$ $1$ .

The volume of the representative washer is

$\int_{0}^{1} π ({(2 - y^{2})}^{2} - {(2 - y)}^{2}) d y = π \int_{0}^{1} (y^{4} - 5 y^{2} + 4 y) d y$ $int_0^1 pi((2-y^2)^2-(2-y)^2) dy = pi int_0^1 (y^4-5y^2+4y) dy$

evaluate to get

$= π (\frac{8}{15}) = \frac{8 π}{15}$ $= pi (8/15) = (8pi)/15$

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How do you find the volume of the solid obtained by rotating the region bounded by the curves $x = y$ $x=y$ and $y = \sqrt{x}$ $y=sqrtx$ about the line $x = 2$ $x=2$ ?

1 Answer

Explanation:

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How do you find the volume of the solid obtained by rotating the region bounded by the curves x=yx=y and y=√xy=sqrtx about the line x=2x=2?

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How do you find the volume of the solid obtained by rotating the region bounded by the curves $x = y$ $x=y$ and $y = \sqrt{x}$ $y=sqrtx$ about the line $x = 2$ $x=2$ ?