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Let the region $R$ be the area enclosed by the function $f(x)=x^{\frac{1}{3}}$ and $g(x)=\frac{1}{2}x$ . Find the volume of the solid generated when the region $R$ is revolved about the line $y=4$ . You may use a calculator and round to the nearest thousandth.

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Find the magnitude and direction of a vector sum

Full solution

Q. Let the region

R

be the area enclosed by the function

f(x)=x^{\frac{1}{3}}

and

g(x)=\frac{1}{2}x

. Find the volume of the solid generated when the region

R

is revolved about the line

y=4

. You may use a calculator and round to the nearest thousandth.

Find Intersection Points: First, we need to find the intersection points of $f(x)$ and $g(x)$ to determine the limits of integration. $\newline$ Set $f(x)$ equal to $g(x)$ : $x^{\frac{1}{3}} = \frac{x}{2}$ .
Set Equations Equal: To solve for $x$ , we cube both sides: $(x^{1/3})^3 = (\frac{x}{2})^3$ . This gives us $x = \frac{x^3}{8}$ .

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