# How do you find the volume generated by revolving about the x-axis, the first quadrant region enclosed by the graphs of #y = 9 - x^2# and #y = 9 - 3x# between 0 to 3?

You are trying to find the volume the 3d figure created by revolving this region around the x-axis

We are going to use the washer method.

First we need to determine the bounds

We can do this by setting both equations equal to each other:

Now we can apply the washer method

In this formula f(x) must be greater than g(x) over the bounds of the integral. In our scenario the function that satisfies this is

Now we just need to plug in f(x) and g(x) into the washer method and integrate.

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To find the volume generated by revolving the region enclosed by the graphs of y = 9 - x^2 and y = 9 - 3x about the x-axis in the interval [0, 3], we use the method of cylindrical shells. The volume (V) is given by the integral:

[V = \int_{0}^{3} 2\pi x \cdot (9 - x^2 - (9 - 3x)) , dx]

This simplifies to:

[V = \int_{0}^{3} 2\pi x \cdot (3x - x^2) , dx]

Integrating this expression yields the volume of the solid of revolution.

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