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`Compute the volume of the solid obtained by rotating the region bounded by $y = x^2$, $x = 0$, and $y = 4$ about the y-axis.`

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

To find the volume of the solid obtained by rotating the region bounded by $y = x^2$, $x = 0$, and $y = 4$ about the y-axis, we use the method of cylindrical shells

step 2

The height of each cylindrical shell is given by $y = x^2$. The radius of each shell is $x$

step 3

The volume of a thin shell is given by $2\pi \cdot \text{radius} \cdot \text{height} \cdot \text{thickness}$. Here, the thickness is $dx$

step 4

The volume of the solid is obtained by integrating the volume of the thin shells from $x = 0$ to $x = 2$

step 5

Set up the integral: $\int_{0}^{2} 2\pi x (x^2) \, dx = 2\pi \int_{0}^{2} x^3 \, dx$

step 6

Evaluate the integral: $2\pi \int_{0}^{2} x^3 \, dx = 2\pi \left[ \frac{x^4}{4} \right]_{0}^{2} = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right) = 2\pi \left( \frac{16}{4} \right) = 2\pi \cdot 4 = 8\pi$

Answer

The volume of the solid is $8\pi$ cubic units.

Key Concept

Cylindrical Shells Method

Explanation

The cylindrical shells method is used to find the volume of a solid of revolution by integrating the volume of thin cylindrical shells.

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