Volumn of x=2sqrt y. About y axis

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In summary, the formula for calculating the volume of x=2sqrt y about the y axis is V = π * ∫(2sqrt y)^2 dy, where V represents the volume, π is the mathematical constant pi, and ∫ represents the integral. This formula is unique as it involves calculating the volume of a shape formed by revolving a curve around the y axis, unlike other volume formulas that involve solid shapes. The "2sqrt" in the formula represents the shape of the curve being revolved and can be used in various real-world applications, such as calculating the volume of water tanks or curved structures. Additionally, the formula can be modified for different shapes by changing the function being revolved around the y axis.
  • #1
karush
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Find volume bounded by
$$x=2\sqrt{y},\ \ x=0, \ \ y=9 $$
About the y axis

So..

$$\pi\int_{0}^{9}(2\sqrt{y})^2 \,dy = 160\pi$$

But the book answer is $162\pi$ ?
 
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  • #2
The book is correct...you have:

\(\displaystyle V=4\pi\int_0^9 y\,dy=2\pi\left(9^2-0^2\right)=162\pi\)
 
  • #3
OK, I see it.
 

FAQ: Volumn of x=2sqrt y. About y axis

What is the formula for calculating the volume of x=2sqrt y about the y axis?

The formula for calculating the volume of x=2sqrt y about the y axis is V = π * ∫(2sqrt y)^2 dy, where V represents the volume, π is the mathematical constant pi, and ∫ represents the integral.

How is the volume of x=2sqrt y about the y axis different from other volume formulas?

The volume of x=2sqrt y about the y axis is different from other volume formulas because it involves calculating the volume of a shape that is formed by revolving a curve around the y axis. This is different from other volume formulas which usually involve calculating the volume of a solid shape with a defined base and height.

What is the significance of the "2sqrt" in the formula for the volume of x=2sqrt y about the y axis?

The "2sqrt" in the formula represents the shape of the curve that is being revolved around the y axis. In this case, it is a half parabola, with the square root function creating a smooth curve. This shape is important in calculating the volume as it determines the cross-sectional area of the shape that is being revolved.

How can I use the volume of x=2sqrt y about the y axis in real-world applications?

The volume of x=2sqrt y about the y axis can be used in various real-world applications, such as calculating the volume of a water tank or a cylindrical container with a curved side. It can also be used in engineering and architecture to calculate the volume of curved structures or objects.

Can the formula for the volume of x=2sqrt y about the y axis be modified for different shapes?

Yes, the formula for the volume of x=2sqrt y about the y axis can be modified for different shapes by changing the function that is being revolved around the y axis. For example, instead of a half parabola, a circle or a different curve can be used, and the formula will be adjusted accordingly.

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