How do you find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=x^2, y=1, about y=2?

Answer 1

See below.

First we find the volume of the area A+B and then subtract the volume of the area B to give us the volume of the area A:

Volume A+B:

It can be see from the diagram that the radius cd of volume A+B is #2-x^2# so the integral will be:

#V=pi*int_(-1)^(1)(2-x^2)^2 dx#

#(2-x^2)^2=4-4x^2+x^4#

#V=pi*int_(-1)^(1)(4-4x^2+x^4) dx=[4x-4/3x^3+1/5x^5]_(-1)^(1)#

#[4x-4/3x^3+1/5x^5]^(1)-[4x-4/3x^3+1/5x^5]_(-1)#

Plugging in upper and lower bounds:

#V=pi*[4(1)-4/3(1)^3+1/5(1)^5]^(1)-[4(-1)-4/3(-1)^3+1/5(-1)^5]_(-1)#

#V=pi*[4-4/3+1/5]^(1)-[-4+4/3-1/5]_(-1)#

#V=pi*[43/15]^(1)-[-43/15]_(-1)=86/15pi#

Volume of B:

This produces a cylinder of radius ab= 1 and length ( this is the length of the interval #[ -1 , 1 ]# which is 2:

#V=pi(1)^2(2)=2pi#

Volume of A= volume(A+B)- volumeB =#(86pi)/15-2pi=color(blue)((56pi)/15)# units cubed.

Volume of A:

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Answer 2

To find the volume of the solid obtained by rotating the region bounded by the curves (y = x^2), (y = 1), and (y = 2) about the axis (y = 2), we can use the disk method.

The region is bounded by (y = x^2) and (y = 1), and it lies between (x = -1) and (x = 1).

The volume of the solid can be calculated as:

[V = \pi \int_{-1}^{1} (2 - x^2)^2 - (2 - 1)^2 , dx]

Simplifying this integral gives:

[V = \pi \int_{-1}^{1} (4 - 4x^2 + x^4) - 1 , dx]

[V = \pi \int_{-1}^{1} 3 - 4x^2 + x^4 , dx]

[V = \pi \left[3x - \frac{4}{3}x^3 + \frac{1}{5}x^5\right]_{-1}^{1}]

[V = \pi \left[(3 - \frac{4}{3} + \frac{1}{5}) - (-3 + \frac{4}{3} - \frac{1}{5})\right]]

[V = \pi \left[\frac{14}{15} + \frac{14}{15}\right]]

[V = \pi \cdot \frac{28}{15}]

So, the volume of the solid is (\frac{28\pi}{15}) cubic units.

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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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