Using the disk method, how do you find the volume of the solid generated by revolving about the xaxis the area bounded by the curves #x=0#, #y=0# and #y=2x+2#?
Looking at the graph of
Next, imagine looking at a crosssection parallel to the xz axis  parallel to the circular base. Every crosssection will be a circle. At
To find the volume of the solid of revolution, we can imagine that our solid is composed of infinitely many disks, of infinitesimal width, and radius equal to
The general formula for the method of disks is:
where
So, in our case, we'll note that
Well, that was the difficult part; setting up the integral. From here, evaluating the integral should be fairly easy. I'll leave it to you as an exercise, but the answer should come out to
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To find the volume of the solid generated by revolving the area bounded by the curves (x=0), (y=0), and (y=2x+2) about the xaxis using the disk method, follow these steps:

Determine the limits of integration by finding the intersection points of the curves. Set (y = 2x + 2) equal to (y = 0) to find the upper limit of integration. [0 = 2x + 2] [x = 1]

The lower limit of integration is (x = 0), as given.

The radius of each disk is the distance from the curve to the axis of revolution. In this case, it is the ycoordinate of the curve. So, the radius (r) is given by (r = y).

The infinitesimal volume (dV) of each disk is calculated using the formula for the volume of a cylinder, (dV = \pi r^2 dx).

Integrate (dV) from the lower limit of integration ((x = 0)) to the upper limit ((x = 1)) to find the total volume: [V = \int_{0}^{1} \pi (2x + 2)^2 , dx]

Simplify and solve the integral to find the volume of the solid.
Thus, by integrating the expression ( \pi (2x + 2)^2 ) from 0 to 1, the volume of the solid generated by revolving the given area about the xaxis can be determined using the disk method.
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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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