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?

Answer 1

#approx 152.681#

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:

#9-x^2=9-3x#
#0=x^2-3x#
#0=x(x-3)#
#x=0,3#

Now we can apply the washer method

#V = pi\int_0^3(f(x)^2-g(x)^2)dx#

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 #y=9-x^2#.

#therefore f(x) = 9-x^2, g(x) = 9-3x#

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

#V = pi\int_0^3((9-x^2)^2-(9-3x)^2)dx#
#=pi\int_0^3((81-18x^2+x^4)-(81-54x+9x^2))dx#
#=pi\int_0^3(x^4-27x^2+54x)dx#
#=pi[x^5/5-9x^3+27x^2]_0^3#
#approx 152.681#

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Answer 2

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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Answer from HIX Tutor

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.

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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