How do you find the volume bounded by #y=sqrtx# and the lines y=0 and x=4 revolved about the y=1?
the volume of a small element is given by
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To find the volume bounded by ( y = \sqrt{x} ) and the lines ( y = 0 ) and ( x = 4 ) revolved about the line ( y = 1 ), you use the method of cylindrical shells.

Determine the limits of integration. Since the region is revolved around the line ( y = 1 ), and the boundaries are ( y = 0 ) and ( x = 4 ), the limits of integration for ( y ) are ( 0 ) to ( 2 ) (since ( y = \sqrt{x} ) intersects ( x = 4 ) at ( y = 2 )).

Set up the integral. The volume ( V ) is given by ( V = \int_{a}^{b} 2\pi rh , dy ), where ( r ) is the distance from the axis of revolution to the curve, and ( h ) is the height of the shell.

Express ( r ) and ( h ) in terms of ( y ). Since we're revolving around the line ( y = 1 ), ( r = y + 1 ) and ( h = 4  y^2 ) (distance between ( x = 4 ) and ( y = \sqrt{x} )).

Substitute ( r ) and ( h ) into the integral and integrate with respect to ( y ) from ( 0 ) to ( 2 ):
[ V = \int_{0}^{2} 2\pi(y + 1)(4  y^2) , dy ]
 Compute the integral to find the volume.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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