question let the region r be the area enclosed by the function f(x)=x^(1/3)-2 and g(x)=(1/2)x - 2. if the…

Explanation:

Step1: Find the intersection points

Set $f(x)=g(x)$, so $x^{\frac{1}{3}} - 2=\frac{1}{2}x - 2$. Then $x^{\frac{1}{3}}=\frac{1}{2}x$. Let $y = x^{\frac{1}{3}}$, the equation becomes $y=\frac{1}{2}y^{3}$, or $y^{3}-2y = 0$, $y(y^{2}-2)=0$. So $y = 0,\pm\sqrt{2}$, and $x = 0,\pm2\sqrt{2}$. The intersection - points are $x = 0$ and $x = 2\sqrt{2}$ (we consider the non - negative intersection point in the domain shown in the graph).

Step2: Find the diameter of the semi - circle

The diameter $d$ of each semi - circle perpendicular to the $x$ - axis is $d=f(x)-g(x)=x^{\frac{1}{3}}-\frac{1}{2}x$.

Step3: Find the radius of the semi - circle

The radius $r=\frac{1}{2}(x^{\frac{1}{3}}-\frac{1}{2}x)$.

Step4: Find the area of the semi - circle

The area of a semi - circle $A=\frac{1}{2}\pi r^{2}=\frac{\pi}{8}(x^{\frac{1}{3}}-\frac{1}{2}x)^{2}=\frac{\pi}{8}(x^{\frac{2}{3}}-x^{\frac{4}{3}}+\frac{1}{4}x^{2})$.

Step5: Find the volume using the integral

The volume $V=\int_{a}^{b}A(x)dx$, where $a = 0$ and $b = 2\sqrt{2}$. So $V=\int_{0}^{2\sqrt{2}}\frac{\pi}{8}(x^{\frac{2}{3}}-x^{\frac{4}{3}}+\frac{1}{4}x^{2})dx$. [ \begin{align*} V&=\frac{\pi}{8}\left[\frac{3}{5}x^{\frac{5}{3}}-\frac{3}{7}x^{\frac{7}{3}}+\frac{1}{12}x^{3}\right]_{0}^{2\sqrt{2}}\ &=\frac{\pi}{8}\left(\frac{3}{5}(2\sqrt{2})^{\frac{5}{3}}-\frac{3}{7}(2\sqrt{2})^{\frac{7}{3}}+\frac{1}{12}(2\sqrt{2})^{3}\right)\ &=\frac{\pi}{8}\left(\frac{3}{5}(2^{\frac{3}{2}})^{\frac{5}{3}}-\frac{3}{7}(2^{\frac{3}{2}})^{\frac{7}{3}}+\frac{1}{12}(2^{\frac{3}{2}})^{2}\right)\ &=\frac{\pi}{8}\left(\frac{3}{5}\times2^{\frac{5}{2}}-\frac{3}{7}\times2^{\frac{7}{2}}+\frac{1}{12}\times8\right)\ &=\frac{\pi}{8}\left(\frac{3}{5}\times4\sqrt{2}-\frac{3}{7}\times8\sqrt{2}+\frac{2}{3}\right)\ &=\frac{\pi}{8}\left(\left(\frac{12\sqrt{2}}{5}-\frac{24\sqrt{2}}{7}\right)+\frac{2}{3}\right)\ &=\frac{\pi}{8}\left(\frac{84\sqrt{2}-120\sqrt{2}}{35}+\frac{2}{3}\right)\ &=\frac{\pi}{8}\left(\frac{- 36\sqrt{2}}{35}+\frac{2}{3}\right)\ &\approx0.196 \end{align*} ]

Answer:

$0.196$