Question

Math

Posted 3 months ago

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Region $R$ is enclosed by the curve $y=\frac{1}{2} x^{2}$ and the line $y=2 x$.
What is the volume of the solid generated when $R$ is rotated about the $y$ axis?
Give an exact answer in terms of $\pi$.
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

To find the volume of the solid generated when region $R$ is rotated about the $y$-axis, we use the method of cylindrical shells. The volume $V$ is given by the integral formula:
$V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx$
where $f(x)$ is the height of the shell and $x$ is the radius

step 2

The region $R$ is enclosed by the curves $y = \frac{1}{2}x^2$ and $y = 2x$. We need to find the points of intersection to determine the limits of integration. Setting $\frac{1}{2}x^2 = 2x$, we solve for $x$:
$\frac{1}{2}x^2 = 2x$
$x^2 = 4x$
$x(x - 4) = 0$
Thus, $x = 0$ and $x = 4$

step 3

The volume integral becomes:
$V = 2\pi \int_{0}^{4} x \left(2x - \frac{1}{2}x^2\right) \, dx$
Simplifying the integrand:
$V = 2\pi \int_{0}^{4} (2x^2 - \frac{1}{2}x^3) \, dx$

step 4

We now integrate term by term:
$V = 2\pi \left[ \int_{0}^{4} 2x^2 \, dx - \int_{0}^{4} \frac{1}{2}x^3 \, dx \right]$
$V = 2\pi \left[ 2 \int_{0}^{4} x^2 \, dx - \frac{1}{2} \int_{0}^{4} x^3 \, dx \right]$

step 5

Compute the integrals:
$\int_{0}^{4} x^2 \, dx = \left. \frac{x^3}{3} \right|_{0}^{4} = \frac{4^3}{3} - \frac{0^3}{3} = \frac{64}{3}$
$\int_{0}^{4} x^3 \, dx = \left. \frac{x^4}{4} \right|_{0}^{4} = \frac{4^4}{4} - \frac{0^4}{4} = 64$

step 6

Substitute back into the volume formula:
$V = 2\pi \left[ 2 \cdot \frac{64}{3} - \frac{1}{2} \cdot 64 \right]$
$V = 2\pi \left[ \frac{128}{3} - 32 \right]$
$V = 2\pi \left[ \frac{128}{3} - \frac{96}{3} \right]$
$V = 2\pi \cdot \frac{32}{3}$
$V = \frac{64\pi}{3}$

Answer

The exact volume of the solid generated when region $R$ is rotated about the $y$-axis is $\frac{64\pi}{3}$.

Key Concept

Volume of solid of revolution using cylindrical shells

Explanation

The volume is calculated by integrating the product of the radius and height of cylindrical shells formed by rotating the region around the $y$-axis.

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