# How do you find the volume of the solid generated by revolving the region bounded by the curves #y^2 = x*(4-x)^2# from x=0 to x=4 rotated about the x-axis?

The answer is

This problem involves finding the volume of a solid of revolution about a disk. Don't let the

However, because the process of finding the volume of a solid of revolution around the x-axis involves *squaring* your expression, it won't matter if you use the positive or negative version. So we'll use the positive.

Step 1. Sketch the region that is going to be revolved and label your boundaries. Also, because the expression in *thickness* rectangle

Step 2. Sketch the 3D solid generated by the region under the curve and the disc formed by the

Step 3. It should be clear that the radius is really just the equation *area of a circle* is *perimeter* of a circle,

Step 4. Find the volume of the solid by integrating over

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To find the volume of the solid generated by revolving the region bounded by the curves (y^2 = x(4-x)^2) from (x=0) to (x=4) rotated about the x-axis, we can use the method of cylindrical shells.

We'll integrate the volume element (dV) along the x-axis from (x=0) to (x=4). The volume of each cylindrical shell is given by (dV = 2\pi y(x) dx), where (y(x)) represents the height of the shell at each point along the x-axis.

First, we need to express (y) in terms of (x). From the given equation (y^2 = x(4-x)^2), we can solve for (y) to get (y = \pm x\sqrt{4-x}). However, since the region is symmetric about the x-axis, we only need to consider the positive values of (y). So, (y = x\sqrt{4-x}).

Next, we integrate (dV) from (x=0) to (x=4): [ V = \int_{0}^{4} 2\pi x\sqrt{4-x} , dx ]

This integral can be evaluated using various techniques, such as substitution or integration by parts. Once the integral is computed, it will give the volume of the solid generated by revolving the region bounded by the given curves about the x-axis.

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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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- Find the length of the curve defined by #y=18(4x^2−2ln(x)), x in[4,6]#?
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- Let R be the region enclosed by #y= e^(2x), y=0, and y=2#. What is the volume of the solid produced by revolving R around the x-axis?

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