# How do you find the volume of the solid obtained by rotating the region bounded by the curves #y^2=4x#, #x=0# and #y=4# about the y axis?

The graph (before rotation) is shown below graph{sqrt(4x) [-0.5, 4.5, -0.5, 4.5]}

Each slice has volume of

The total volume is given by

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To find the volume of the solid obtained by rotating the region bounded by the curves ( y^2 = 4x ), ( x = 0 ), and ( y = 4 ) about the y-axis, you can use the method of cylindrical shells.

The formula to find the volume using cylindrical shells is:

[ V = 2\pi \int_a^b xf(x) , dx ]

In this case, the bounds of integration are from ( x = 0 ) to ( x = 4 ), as these are the x-coordinates where the curves intersect.

First, solve ( y^2 = 4x ) for ( x ) to get ( x = \frac{y^2}{4} ). Then, the equation ( x = 4 ) represents the vertical line.

Now, the integral becomes:

[ V = 2\pi \int_0^4 x\sqrt{4x} , dx ]

Simplify the integral:

[ V = 2\pi \int_0^4 2x^{3/2} , dx ]

[ V = 4\pi \int_0^4 x^{3/2} , dx ]

Integrate ( x^{3/2} ) with respect to ( x ):

[ V = 4\pi \left[ \frac{2}{5} x^{5/2} \right]_0^4 ]

[ V = \frac{8\pi}{5} (4^{5/2} - 0) ]

[ V = \frac{8\pi}{5} (32) ]

[ V = \frac{256\pi}{5} ]

So, the volume of the solid obtained by rotating the region bounded by the curves ( y^2 = 4x ), ( x = 0 ), and ( y = 4 ) about the y-axis is ( \frac{256\pi}{5} ).

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When evaluating a one-sided 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 one-sided 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 one-sided 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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