`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 5 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 a typical shell is given by $h = y = x^2$

step 3

The radius of a typical shell is $r = x$

step 4

The volume of a typical shell is $dV = 2 \pi x \cdot h \cdot dx = 2 \pi x \cdot x^2 \cdot dx = 2 \pi x^3 \cdot dx$

step 5

To find the total volume, we integrate $dV$ from $x = 0$ to $x = 2$ (since $y = 4$ corresponds to $x = 2$):

step 6

$V = \int_{0}^{2} 2 \pi x^3 \, dx$

step 7

Compute the integral:

step 8

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

Answer

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

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 infinitesimally thin cylindrical shells.

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