# The first quadrant region enclosed by y=2x, the x-axis and the line x=1 is resolved about the line y=0. How do you find the resulting volume?

Evaluate

The region is shown in the graph below.

The graph shows a representative slice of the region. The slice is taken at some value of

When rotated about the

The volume of the representative disk is

In this case we have:

the volume of the slice is

Note

Because the question was posted under the topic "Calculating Volumes using Integrals", I have used the integral.

With some imagination we might notice that the solid will be a right circular cone.

The cone will have radius

The volume of a right circular cone is

Using this formula we also get

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To find the resulting volume, you can use the method of cylindrical shells. First, determine the limits of integration by finding the intersection point between y = 2x and x = 1, which is (1, 2). Then, the radius of each shell is the distance from the line of rotation (y = 0) to the axis of rotation (x = 1), which is 1. The height of each shell is given by the function y = 2x. Therefore, the integral for the volume is from x = 0 to x = 1 of 2πx(1)dx. Integrate this expression to find the volume.

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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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